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Properties

Label 2-1700-1700.1183-c0-0-0
Degree $2$
Conductor $1700$
Sign $0.0989 - 0.995i$
Analytic cond. $0.848410$
Root an. cond. $0.921092$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.760 + 0.649i)2-s + (0.156 − 0.987i)4-s + (0.852 − 0.522i)5-s + (0.522 + 0.852i)8-s + (0.0784 + 0.996i)9-s + (−0.309 + 0.951i)10-s + (−0.600 + 1.84i)13-s + (−0.951 − 0.309i)16-s + (−0.156 + 0.987i)17-s + (−0.707 − 0.707i)18-s + (−0.382 − 0.923i)20-s + (0.453 − 0.891i)25-s + (−0.744 − 1.79i)26-s + (−1.18 + 1.28i)29-s + (0.923 − 0.382i)32-s + ⋯
L(s)  = 1  + (−0.760 + 0.649i)2-s + (0.156 − 0.987i)4-s + (0.852 − 0.522i)5-s + (0.522 + 0.852i)8-s + (0.0784 + 0.996i)9-s + (−0.309 + 0.951i)10-s + (−0.600 + 1.84i)13-s + (−0.951 − 0.309i)16-s + (−0.156 + 0.987i)17-s + (−0.707 − 0.707i)18-s + (−0.382 − 0.923i)20-s + (0.453 − 0.891i)25-s + (−0.744 − 1.79i)26-s + (−1.18 + 1.28i)29-s + (0.923 − 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0989 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0989 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.0989 - 0.995i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 0.0989 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8346306289\)
\(L(\frac12)\) \(\approx\) \(0.8346306289\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.760 - 0.649i)T \)
5 \( 1 + (-0.852 + 0.522i)T \)
17 \( 1 + (0.156 - 0.987i)T \)
good3 \( 1 + (-0.0784 - 0.996i)T^{2} \)
7 \( 1 + (-0.382 + 0.923i)T^{2} \)
11 \( 1 + (0.233 + 0.972i)T^{2} \)
13 \( 1 + (0.600 - 1.84i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.453 - 0.891i)T^{2} \)
23 \( 1 + (0.972 - 0.233i)T^{2} \)
29 \( 1 + (1.18 - 1.28i)T + (-0.0784 - 0.996i)T^{2} \)
31 \( 1 + (-0.760 + 0.649i)T^{2} \)
37 \( 1 + (0.145 - 1.22i)T + (-0.972 - 0.233i)T^{2} \)
41 \( 1 + (-0.663 + 1.18i)T + (-0.522 - 0.852i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.891 + 1.45i)T + (-0.453 - 0.891i)T^{2} \)
59 \( 1 + (-0.987 + 0.156i)T^{2} \)
61 \( 1 + (-1.98 + 0.234i)T + (0.972 - 0.233i)T^{2} \)
67 \( 1 + (0.951 - 0.309i)T^{2} \)
71 \( 1 + (-0.996 + 0.0784i)T^{2} \)
73 \( 1 + (0.187 - 0.666i)T + (-0.852 - 0.522i)T^{2} \)
79 \( 1 + (0.760 + 0.649i)T^{2} \)
83 \( 1 + (0.891 + 0.453i)T^{2} \)
89 \( 1 + (1.77 + 0.905i)T + (0.587 + 0.809i)T^{2} \)
97 \( 1 + (-0.234 + 0.00922i)T + (0.996 - 0.0784i)T^{2} \)
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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

−9.657931656778403800644040519920, −8.775881129043272059703028849386, −8.411222235047769924845185583041, −7.18900353722314386268836302029, −6.75658763961190068080703463994, −5.67766460121723035234461549494, −5.07620492330351422854582917710, −4.17616582917808416765302284711, −2.16842465645264078523221705535, −1.68075929557754102923093665717, 0.838097898000839788912044359396, 2.39858176529485698705225371394, 2.99321184676570461681633288585, 4.02970206903258432931838778661, 5.40497003435683365738507703244, 6.18238081874973416932768893945, 7.24350959555173784727267388368, 7.70235464598546733940428608763, 8.835258071719701199165694464251, 9.577546741433850599656367986729

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