Properties

Label 4-1040e2-1.1-c1e2-0-41
Degree $4$
Conductor $1081600$
Sign $1$
Analytic cond. $68.9637$
Root an. cond. $2.88174$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 4·9-s − 4·13-s + 4·17-s − 25-s − 2·29-s − 4·37-s + 12·41-s + 8·45-s + 10·49-s + 10·53-s + 10·61-s − 8·65-s + 7·81-s + 8·85-s + 16·89-s − 26·97-s + 20·101-s − 6·109-s + 10·113-s − 16·117-s + 10·121-s − 12·125-s + ⋯
L(s)  = 1  + 0.894·5-s + 4/3·9-s − 1.10·13-s + 0.970·17-s − 1/5·25-s − 0.371·29-s − 0.657·37-s + 1.87·41-s + 1.19·45-s + 10/7·49-s + 1.37·53-s + 1.28·61-s − 0.992·65-s + 7/9·81-s + 0.867·85-s + 1.69·89-s − 2.63·97-s + 1.99·101-s − 0.574·109-s + 0.940·113-s − 1.47·117-s + 0.909·121-s − 1.07·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(68.9637\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.874927842\)
\(L(\frac12)\) \(\approx\) \(2.874927842\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.23.a_aba
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.c_bi
31$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.31.a_u
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.am_dy
43$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.43.a_aq
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.53.ak_fa
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.59.a_cs
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) 2.61.ak_es
67$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.67.a_aby
71$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.71.a_abo
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.83.a_dm
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.aq_je
97$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.ba_na
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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

−7.930463874828633825062224737317, −7.53003567323360384855179850359, −7.21268304790497315794224976103, −6.94543826347514924714413199834, −6.23492860894477625561592166633, −5.87249983102043535773372573778, −5.37100574636528683528211030426, −5.05217345650030833540225063885, −4.40968645782809549518775147425, −3.95537218007162388619672483448, −3.50252919317468199258507794397, −2.53738759831615501237901022320, −2.30076986848220875955056739552, −1.54170811786572283937180225573, −0.814072322244636703764266444816, 0.814072322244636703764266444816, 1.54170811786572283937180225573, 2.30076986848220875955056739552, 2.53738759831615501237901022320, 3.50252919317468199258507794397, 3.95537218007162388619672483448, 4.40968645782809549518775147425, 5.05217345650030833540225063885, 5.37100574636528683528211030426, 5.87249983102043535773372573778, 6.23492860894477625561592166633, 6.94543826347514924714413199834, 7.21268304790497315794224976103, 7.53003567323360384855179850359, 7.930463874828633825062224737317

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