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Properties

Label 4-26e2-1.1-c1e2-0-0
Degree $4$
Conductor $676$
Sign $1$
Analytic cond. $0.0431023$
Root an. cond. $0.455643$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·5-s − 4·7-s + 8-s + 3·9-s + 2·10-s − 4·11-s + 7·13-s + 4·14-s − 16-s − 3·17-s − 3·18-s + 4·22-s + 4·23-s − 7·25-s − 7·26-s + 29-s + 8·31-s + 3·34-s + 8·35-s − 3·37-s − 2·40-s + 9·41-s + 8·43-s − 6·45-s − 4·46-s − 16·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.894·5-s − 1.51·7-s + 0.353·8-s + 9-s + 0.632·10-s − 1.20·11-s + 1.94·13-s + 1.06·14-s − 1/4·16-s − 0.727·17-s − 0.707·18-s + 0.852·22-s + 0.834·23-s − 7/5·25-s − 1.37·26-s + 0.185·29-s + 1.43·31-s + 0.514·34-s + 1.35·35-s − 0.493·37-s − 0.316·40-s + 1.40·41-s + 1.21·43-s − 0.894·45-s − 0.589·46-s − 2.33·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.0431023\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 676,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2888268393\)
\(L(\frac12)\) \(\approx\) \(0.2888268393\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2$C_2$ \( 1 + T + T^{2} \)
13$C_2$ \( 1 - 7 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.a_ad
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.5.c_l
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.e_j
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_f
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.19.a_at
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) 2.29.ab_abc
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.37.d_abc
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_bo
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.ai_v
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_abr
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_am
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_abz
71$C_2^2$ \( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_ah
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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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

−17.83917270532275816683966581168, −17.50799021433546123665920387934, −16.26623720731594403394103939384, −16.02911031406074142986274222501, −15.64886343391872673852350459186, −15.36446925692976134734598803910, −13.90625455460884407486336769723, −13.30524454945633262033065358911, −12.96336081772086273433631127787, −12.28901214179054392561517743721, −11.07879755217083448900979344953, −10.86528555775159191282929273534, −9.795272204285249139218264653633, −9.455354274309319704863065403827, −8.321664699454448960348307050955, −7.87360442404991356808984330857, −6.81910561654721702742782169449, −6.08899132613771253922925065251, −4.44784784558725094005357709970, −3.37568655586676820268862357978, 3.37568655586676820268862357978, 4.44784784558725094005357709970, 6.08899132613771253922925065251, 6.81910561654721702742782169449, 7.87360442404991356808984330857, 8.321664699454448960348307050955, 9.455354274309319704863065403827, 9.795272204285249139218264653633, 10.86528555775159191282929273534, 11.07879755217083448900979344953, 12.28901214179054392561517743721, 12.96336081772086273433631127787, 13.30524454945633262033065358911, 13.90625455460884407486336769723, 15.36446925692976134734598803910, 15.64886343391872673852350459186, 16.02911031406074142986274222501, 16.26623720731594403394103939384, 17.50799021433546123665920387934, 17.83917270532275816683966581168

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