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Properties

Label 2-1700-68.67-c0-0-6
Degree $2$
Conductor $1700$
Sign $1$
Analytic cond. $0.848410$
Root an. cond. $0.921092$
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 + 4-s + 8-s − 9-s + 2·13-s + 16-s − 17-s − 18-s + 2·26-s + 32-s − 34-s − 36-s − 49-s + 2·52-s − 2·53-s + 64-s − 68-s − 72-s + 81-s − 2·89-s − 98-s − 2·101-s + 2·104-s − 2·106-s − 2·117-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s − 9-s + 2·13-s + 16-s − 17-s − 18-s + 2·26-s + 32-s − 34-s − 36-s − 49-s + 2·52-s − 2·53-s + 64-s − 68-s − 72-s + 81-s − 2·89-s − 98-s − 2·101-s + 2·104-s − 2·106-s − 2·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1700} (951, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.113874648\)
\(L(\frac12)\) \(\approx\) \(2.113874648\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
17 \( 1 + T \)
good3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( ( 1 - T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 + T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 + T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.498844936416581133099689775269, −8.545512767504470141868270146426, −8.027504814865453575327452767829, −6.77974772835202928172017816387, −6.20796791261749871908222624526, −5.54053055292698683506258227511, −4.51884831905380048511814896770, −3.62962237744920929049211856267, −2.83565499262466302597795104399, −1.58618694677452289913428165946, 1.58618694677452289913428165946, 2.83565499262466302597795104399, 3.62962237744920929049211856267, 4.51884831905380048511814896770, 5.54053055292698683506258227511, 6.20796791261749871908222624526, 6.77974772835202928172017816387, 8.027504814865453575327452767829, 8.545512767504470141868270146426, 9.498844936416581133099689775269

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