Properties

Label 2-35-35.34-c4-0-8
Degree $2$
Conductor $35$
Sign $1$
Analytic cond. $3.61794$
Root an. cond. $1.90209$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 17·3-s + 16·4-s − 25·5-s − 49·7-s + 208·9-s − 73·11-s + 272·12-s − 23·13-s − 425·15-s + 256·16-s − 263·17-s − 400·20-s − 833·21-s + 625·25-s + 2.15e3·27-s − 784·28-s − 1.15e3·29-s − 1.24e3·33-s + 1.22e3·35-s + 3.32e3·36-s − 391·39-s − 1.16e3·44-s − 5.20e3·45-s + 3.45e3·47-s + 4.35e3·48-s + 2.40e3·49-s − 4.47e3·51-s + ⋯
L(s)  = 1  + 17/9·3-s + 4-s − 5-s − 7-s + 2.56·9-s − 0.603·11-s + 17/9·12-s − 0.136·13-s − 1.88·15-s + 16-s − 0.910·17-s − 20-s − 1.88·21-s + 25-s + 2.96·27-s − 28-s − 1.37·29-s − 1.13·33-s + 35-s + 2.56·36-s − 0.257·39-s − 0.603·44-s − 2.56·45-s + 1.56·47-s + 17/9·48-s + 49-s − 1.71·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $1$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: $\chi_{35} (34, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.328569801\)
\(L(\frac12)\) \(\approx\) \(2.328569801\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + p^{2} T \)
7 \( 1 + p^{2} T \)
good2 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
3 \( 1 - 17 T + p^{4} T^{2} \)
11 \( 1 + 73 T + p^{4} T^{2} \)
13 \( 1 + 23 T + p^{4} T^{2} \)
17 \( 1 + 263 T + p^{4} T^{2} \)
19 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
23 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
29 \( 1 + 1153 T + p^{4} T^{2} \)
31 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
37 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
41 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
43 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
47 \( 1 - 3457 T + p^{4} T^{2} \)
53 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
59 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
61 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
67 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
71 \( 1 + 10078 T + p^{4} T^{2} \)
73 \( 1 - 9502 T + p^{4} T^{2} \)
79 \( 1 - 12167 T + p^{4} T^{2} \)
83 \( 1 - 6382 T + p^{4} T^{2} \)
89 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
97 \( 1 + 3383 T + p^{4} T^{2} \)
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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

−15.51487124230517903403246716626, −14.94843346995382516901317179392, −13.41378277649849048883335845106, −12.40308441785771029161524693376, −10.65105576750368463917003000635, −9.213682664261132612134472079807, −7.889029152079680423832801040629, −6.97696556564300181827071952644, −3.73081912494297591235812834728, −2.53257142807090831180213907965, 2.53257142807090831180213907965, 3.73081912494297591235812834728, 6.97696556564300181827071952644, 7.889029152079680423832801040629, 9.213682664261132612134472079807, 10.65105576750368463917003000635, 12.40308441785771029161524693376, 13.41378277649849048883335845106, 14.94843346995382516901317179392, 15.51487124230517903403246716626

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