Properties

Label 2-35-35.34-c12-0-25
Degree $2$
Conductor $35$
Sign $1$
Analytic cond. $31.9897$
Root an. cond. $5.65595$
Motivic weight $12$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 782·3-s + 4.09e3·4-s + 1.56e4·5-s + 1.17e5·7-s + 8.00e4·9-s + 2.81e6·11-s − 3.20e6·12-s − 1.95e6·13-s − 1.22e7·15-s + 1.67e7·16-s − 4.77e7·17-s + 6.40e7·20-s − 9.20e7·21-s + 2.44e8·25-s + 3.52e8·27-s + 4.81e8·28-s + 9.13e8·29-s − 2.20e9·33-s + 1.83e9·35-s + 3.28e8·36-s + 1.53e9·39-s + 1.15e10·44-s + 1.25e9·45-s + 9.29e9·47-s − 1.31e10·48-s + 1.38e10·49-s + 3.73e10·51-s + ⋯
L(s)  = 1  − 1.07·3-s + 4-s + 5-s + 7-s + 0.150·9-s + 1.59·11-s − 1.07·12-s − 0.405·13-s − 1.07·15-s + 16-s − 1.97·17-s + 20-s − 1.07·21-s + 25-s + 0.911·27-s + 28-s + 1.53·29-s − 1.70·33-s + 35-s + 0.150·36-s + 0.435·39-s + 1.59·44-s + 0.150·45-s + 0.862·47-s − 1.07·48-s + 49-s + 2.12·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $1$
Analytic conductor: \(31.9897\)
Root analytic conductor: \(5.65595\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: $\chi_{35} (34, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :6),\ 1)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.500669605\)
\(L(\frac12)\) \(\approx\) \(2.500669605\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - p^{6} T \)
7 \( 1 - p^{6} T \)
good2 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
3 \( 1 + 782 T + p^{12} T^{2} \)
11 \( 1 - 2817362 T + p^{12} T^{2} \)
13 \( 1 + 1958542 T + p^{12} T^{2} \)
17 \( 1 + 47706622 T + p^{12} T^{2} \)
19 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
23 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
29 \( 1 - 913676402 T + p^{12} T^{2} \)
31 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
37 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
41 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
43 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
47 \( 1 - 9293086658 T + p^{12} T^{2} \)
53 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
59 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
61 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
67 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
71 \( 1 + 255286231198 T + p^{12} T^{2} \)
73 \( 1 + 48396356062 T + p^{12} T^{2} \)
79 \( 1 - 379435754882 T + p^{12} T^{2} \)
83 \( 1 - 648698638898 T + p^{12} T^{2} \)
89 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
97 \( 1 + 859766289982 T + p^{12} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.94363813031684817198633046069, −12.18789605923239860099719537705, −11.38375173389835821275917100084, −10.52583469427103103819344969890, −8.856446019422324245621854040214, −6.83209085099345128773132635889, −6.10222296140079749432957094605, −4.70356052383468584982235803830, −2.27465426506998635043147479361, −1.11925552110328706654861211915, 1.11925552110328706654861211915, 2.27465426506998635043147479361, 4.70356052383468584982235803830, 6.10222296140079749432957094605, 6.83209085099345128773132635889, 8.856446019422324245621854040214, 10.52583469427103103819344969890, 11.38375173389835821275917100084, 12.18789605923239860099719537705, 13.94363813031684817198633046069

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