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Properties

Label 2-1700-1700.1379-c0-0-0
Degree $2$
Conductor $1700$
Sign $0.548 - 0.836i$
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  + (−0.453 − 0.891i)2-s + (−0.587 + 0.809i)4-s + (−0.987 − 0.156i)5-s + (0.987 + 0.156i)8-s + (−0.891 − 0.453i)9-s + (0.309 + 0.951i)10-s + (0.0966 + 0.297i)13-s + (−0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s + 1.00i·18-s + (0.707 − 0.707i)20-s + (0.951 + 0.309i)25-s + (0.221 − 0.221i)26-s + (−0.465 + 1.93i)29-s + (−0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.453 − 0.891i)2-s + (−0.587 + 0.809i)4-s + (−0.987 − 0.156i)5-s + (0.987 + 0.156i)8-s + (−0.891 − 0.453i)9-s + (0.309 + 0.951i)10-s + (0.0966 + 0.297i)13-s + (−0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s + 1.00i·18-s + (0.707 − 0.707i)20-s + (0.951 + 0.309i)25-s + (0.221 − 0.221i)26-s + (−0.465 + 1.93i)29-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.548 - 0.836i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 0.548 - 0.836i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3519467307\)
\(L(\frac12)\) \(\approx\) \(0.3519467307\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.453 + 0.891i)T \)
5 \( 1 + (0.987 + 0.156i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
good3 \( 1 + (0.891 + 0.453i)T^{2} \)
7 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.156 + 0.987i)T^{2} \)
13 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.951 - 0.309i)T^{2} \)
23 \( 1 + (-0.156 - 0.987i)T^{2} \)
29 \( 1 + (0.465 - 1.93i)T + (-0.891 - 0.453i)T^{2} \)
31 \( 1 + (0.453 + 0.891i)T^{2} \)
37 \( 1 + (0.987 - 0.843i)T + (0.156 - 0.987i)T^{2} \)
41 \( 1 + (-0.465 - 0.0366i)T + (0.987 + 0.156i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (1.95 - 0.309i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (-0.587 - 0.809i)T^{2} \)
61 \( 1 + (-1.47 - 1.26i)T + (0.156 + 0.987i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.891 - 0.453i)T^{2} \)
73 \( 1 + (-0.0819 - 1.04i)T + (-0.987 + 0.156i)T^{2} \)
79 \( 1 + (0.453 - 0.891i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
89 \( 1 + (1.69 + 0.550i)T + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (1.47 + 0.355i)T + (0.891 + 0.453i)T^{2} \)
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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

−9.581226485347804593251011181747, −8.623060959260584885713956278028, −8.577678570120816553634121292314, −7.46838254432190257764656601741, −6.67475812498951288873599306447, −5.39745687481286477607516260676, −4.39249716261579688567406160240, −3.60316739546623307757047060166, −2.83348283141836778375602404847, −1.39952828155611162408715615482, 0.32905139289381597737333822044, 2.31756797552344404347325239924, 3.63564105140008397873237180264, 4.62527979400434484125391586093, 5.41949051344950309480746225655, 6.31255523720394097898431700462, 7.17942795779257080686346366657, 7.87428009663507458000628982795, 8.409888367124078645557109801661, 9.185098844024206049701161742862

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