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Properties

Label 2-1700-1700.1239-c0-0-1
Degree $2$
Conductor $1700$
Sign $-0.965 - 0.261i$
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  + (0.156 − 0.987i)2-s + (−0.951 − 0.309i)4-s + (0.453 − 0.891i)5-s + (−0.453 + 0.891i)8-s + (−0.987 + 0.156i)9-s + (−0.809 − 0.587i)10-s + (−1.44 − 1.04i)13-s + (0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + i·18-s + (−0.707 + 0.707i)20-s + (−0.587 − 0.809i)25-s + (−1.26 + 1.26i)26-s + (−0.0819 − 1.04i)29-s + (0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (0.156 − 0.987i)2-s + (−0.951 − 0.309i)4-s + (0.453 − 0.891i)5-s + (−0.453 + 0.891i)8-s + (−0.987 + 0.156i)9-s + (−0.809 − 0.587i)10-s + (−1.44 − 1.04i)13-s + (0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + i·18-s + (−0.707 + 0.707i)20-s + (−0.587 − 0.809i)25-s + (−1.26 + 1.26i)26-s + (−0.0819 − 1.04i)29-s + (0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.965 - 0.261i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ -0.965 - 0.261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6925118874\)
\(L(\frac12)\) \(\approx\) \(0.6925118874\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 + 0.987i)T \)
5 \( 1 + (-0.453 + 0.891i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
good3 \( 1 + (0.987 - 0.156i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.891 - 0.453i)T^{2} \)
13 \( 1 + (1.44 + 1.04i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.587 + 0.809i)T^{2} \)
23 \( 1 + (-0.891 + 0.453i)T^{2} \)
29 \( 1 + (0.0819 + 1.04i)T + (-0.987 + 0.156i)T^{2} \)
31 \( 1 + (-0.156 + 0.987i)T^{2} \)
37 \( 1 + (-0.453 - 0.108i)T + (0.891 + 0.453i)T^{2} \)
41 \( 1 + (-0.0819 - 0.133i)T + (-0.453 + 0.891i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.412 + 0.809i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.951 + 0.309i)T^{2} \)
61 \( 1 + (-1.93 + 0.465i)T + (0.891 - 0.453i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (-0.987 + 0.156i)T^{2} \)
73 \( 1 + (-1.29 - 0.794i)T + (0.453 + 0.891i)T^{2} \)
79 \( 1 + (-0.156 - 0.987i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (-1.16 - 1.59i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (1.93 - 0.152i)T + (0.987 - 0.156i)T^{2} \)
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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

−9.416474034835083225793130286764, −8.377363414416495843285669402119, −7.970757666728143355207348125654, −6.48926744737033074636607014295, −5.35508454209393385189052060659, −5.10730787484992610551875498792, −4.05502952943983892398903559396, −2.76616329809908972396911609625, −2.14432600264672687498905236610, −0.47025372749124063868146384672, 2.23028156091514542771601116508, 3.21379693743130048130159594572, 4.35103918484994077236980668076, 5.22621571413292504461105217089, 6.09557812215492564269309493535, 6.76998408844528262455298448076, 7.33204206269638320721109802714, 8.306669164293633922889766242390, 9.189305642129969451818951816569, 9.612311343591421331586015629997

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