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Properties

Label 1-35-35.34-r1-0-0
Degree $1$
Conductor $35$
Sign $1$
Analytic cond. $3.76127$
Root an. cond. $3.76127$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 11-s + 12-s + 13-s + 16-s + 17-s − 18-s − 19-s − 22-s − 23-s − 24-s − 26-s + 27-s + 29-s − 31-s − 32-s + 33-s − 34-s + 36-s − 37-s + 38-s + 39-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 11-s + 12-s + 13-s + 16-s + 17-s − 18-s − 19-s − 22-s − 23-s − 24-s − 26-s + 27-s + 29-s − 31-s − 32-s + 33-s − 34-s + 36-s − 37-s + 38-s + 39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $1$
Analytic conductor: \(3.76127\)
Root analytic conductor: \(3.76127\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{35} (34, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 35,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.372465237\)
\(L(\frac12)\) \(\approx\) \(1.372465237\)
\(L(1)\) \(\approx\) \(1.062052159\)
\(L(1)\) \(\approx\) \(1.062052159\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−35.93656709262921494800829015810, −34.81663022808999437230757120258, −33.29775946630843841084415985314, −32.21329758026752415147926390349, −30.53767072864195136131283098629, −29.77591034704889739340766074297, −28.034897209026994405575809826739, −27.14893274813990933298817921517, −25.78369694566814558166376696303, −25.1908516827122906003733363186, −23.767962899232326681229391300300, −21.54249566958256605788791652266, −20.39564929846450604411390726204, −19.359777176426702689199484565686, −18.30931008174258640414742456185, −16.71535223511675595666894858584, −15.39842379771834236904095193693, −14.09506805533827518506273772297, −12.215948939308714465948507514834, −10.453275271443690457135814505185, −9.11209558517578644537972038357, −8.07013521466856548869832839375, −6.50832203154858485335713838627, −3.54730149273500900690953691213, −1.62321411858135276372325616243, 1.62321411858135276372325616243, 3.54730149273500900690953691213, 6.50832203154858485335713838627, 8.07013521466856548869832839375, 9.11209558517578644537972038357, 10.453275271443690457135814505185, 12.215948939308714465948507514834, 14.09506805533827518506273772297, 15.39842379771834236904095193693, 16.71535223511675595666894858584, 18.30931008174258640414742456185, 19.359777176426702689199484565686, 20.39564929846450604411390726204, 21.54249566958256605788791652266, 23.767962899232326681229391300300, 25.1908516827122906003733363186, 25.78369694566814558166376696303, 27.14893274813990933298817921517, 28.034897209026994405575809826739, 29.77591034704889739340766074297, 30.53767072864195136131283098629, 32.21329758026752415147926390349, 33.29775946630843841084415985314, 34.81663022808999437230757120258, 35.93656709262921494800829015810

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