Properties

Label 2-35-1.1-c1-0-2
Degree $2$
Conductor $35$
Sign $1$
Analytic cond. $0.279476$
Root an. cond. $0.528655$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·2-s − 2.56·3-s + 0.438·4-s + 5-s − 4·6-s − 7-s − 2.43·8-s + 3.56·9-s + 1.56·10-s + 2.56·11-s − 1.12·12-s + 4.56·13-s − 1.56·14-s − 2.56·15-s − 4.68·16-s − 4.56·17-s + 5.56·18-s + 1.12·19-s + 0.438·20-s + 2.56·21-s + 4·22-s − 5.12·23-s + 6.24·24-s + 25-s + 7.12·26-s − 1.43·27-s − 0.438·28-s + ⋯
L(s)  = 1  + 1.10·2-s − 1.47·3-s + 0.219·4-s + 0.447·5-s − 1.63·6-s − 0.377·7-s − 0.862·8-s + 1.18·9-s + 0.493·10-s + 0.772·11-s − 0.324·12-s + 1.26·13-s − 0.417·14-s − 0.661·15-s − 1.17·16-s − 1.10·17-s + 1.31·18-s + 0.257·19-s + 0.0980·20-s + 0.558·21-s + 0.852·22-s − 1.06·23-s + 1.27·24-s + 0.200·25-s + 1.39·26-s − 0.276·27-s − 0.0828·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $1$
Analytic conductor: \(0.279476\)
Root analytic conductor: \(0.528655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8101846184\)
\(L(\frac12)\) \(\approx\) \(0.8101846184\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 + T \)
good2 \( 1 - 1.56T + 2T^{2} \)
3 \( 1 + 2.56T + 3T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 + 4.56T + 17T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 + 5.12T + 23T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 3.12T + 41T^{2} \)
43 \( 1 - 9.12T + 43T^{2} \)
47 \( 1 - 3.68T + 47T^{2} \)
53 \( 1 - 3.12T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 9.36T + 61T^{2} \)
67 \( 1 + 6.24T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 + 6.56T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 7.12T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−16.54043015310824634377959576393, −15.47778975023287695392669568729, −13.94588673316197709173832397868, −12.99212700705068834617634779508, −11.90227741643907382762501423525, −10.91335028499141244772470576035, −9.212565164174387439717095614639, −6.44771243960593321795590797896, −5.75333263909241173971323500780, −4.15686477265978088320503087188, 4.15686477265978088320503087188, 5.75333263909241173971323500780, 6.44771243960593321795590797896, 9.212565164174387439717095614639, 10.91335028499141244772470576035, 11.90227741643907382762501423525, 12.99212700705068834617634779508, 13.94588673316197709173832397868, 15.47778975023287695392669568729, 16.54043015310824634377959576393

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