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Properties

Label 4-13e2-1.1-c1e2-0-0
Degree $4$
Conductor $169$
Sign $1$
Analytic cond. $0.0107755$
Root an. cond. $0.322188$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 2·3-s + 4·4-s + 6·6-s − 3·8-s + 3·9-s − 8·12-s − 5·13-s + 3·16-s + 3·17-s − 9·18-s − 6·19-s + 6·23-s + 6·24-s + 7·25-s + 15·26-s − 10·27-s − 3·29-s − 6·32-s − 9·34-s + 12·36-s + 15·37-s + 18·38-s + 10·39-s − 9·41-s − 8·43-s − 18·46-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 2·4-s + 2.44·6-s − 1.06·8-s + 9-s − 2.30·12-s − 1.38·13-s + 3/4·16-s + 0.727·17-s − 2.12·18-s − 1.37·19-s + 1.25·23-s + 1.22·24-s + 7/5·25-s + 2.94·26-s − 1.92·27-s − 0.557·29-s − 1.06·32-s − 1.54·34-s + 2·36-s + 2.46·37-s + 2.91·38-s + 1.60·39-s − 1.40·41-s − 1.21·43-s − 2.65·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $1$
Analytic conductor: \(0.0107755\)
Root analytic conductor: \(0.322188\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 169,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09049039083\)
\(L(\frac12)\) \(\approx\) \(0.09049039083\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad13$C_2$ \( 1 + 5 T + p T^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.2.d_f
3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_b
5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.5.a_ah
7$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.7.a_h
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.11.a_l
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.g_bf
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_au
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2^2$ \( 1 - 15 T + 112 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.37.ap_ei
41$C_2^2$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_cq
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.47.a_ade
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.53.g_el
59$C_2^2$ \( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_ed
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.b_aci
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.ag_db
71$C_2^2$ \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_df
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.a_afn
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.83.a_ba
89$C_2^2$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_fh
97$C_2^2$ \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_fp
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.10703301806788866096607442711, −19.33846444803067811938292685952, −18.76925999761091455577670097918, −18.56576970884593164181156978722, −17.65895984226676047880460224981, −17.20457702255851851831821238873, −16.64032221486140008944897783369, −16.58357118750575786137645040830, −14.95140792841582142043623560609, −14.90357896669741344337333157041, −13.06339550571964159220878092128, −12.51224222861724419404676510427, −11.45861743544082710207892638661, −10.84883986345867375872722724058, −9.917477007979000496132305323081, −9.565832078364084782726879302558, −8.455526054396895167012626272507, −7.54467569516254146308765020154, −6.57037775431560330867930744613, −5.06823463540602925038855173519, 5.06823463540602925038855173519, 6.57037775431560330867930744613, 7.54467569516254146308765020154, 8.455526054396895167012626272507, 9.565832078364084782726879302558, 9.917477007979000496132305323081, 10.84883986345867375872722724058, 11.45861743544082710207892638661, 12.51224222861724419404676510427, 13.06339550571964159220878092128, 14.90357896669741344337333157041, 14.95140792841582142043623560609, 16.58357118750575786137645040830, 16.64032221486140008944897783369, 17.20457702255851851831821238873, 17.65895984226676047880460224981, 18.56576970884593164181156978722, 18.76925999761091455577670097918, 19.33846444803067811938292685952, 20.10703301806788866096607442711

Graph of the $Z$-function along the critical line