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Properties

Label 2-1700-1700.1271-c0-0-1
Degree $2$
Conductor $1700$
Sign $0.862 - 0.506i$
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  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (0.587 + 0.809i)8-s + (−0.951 + 0.309i)9-s + (0.951 − 0.309i)10-s + (0.363 + 1.11i)13-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s − 0.999·18-s + 0.999·20-s + (0.309 − 0.951i)25-s + 1.17i·26-s + (0.142 + 0.896i)29-s + i·32-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (0.587 + 0.809i)8-s + (−0.951 + 0.309i)9-s + (0.951 − 0.309i)10-s + (0.363 + 1.11i)13-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s − 0.999·18-s + 0.999·20-s + (0.309 − 0.951i)25-s + 1.17i·26-s + (0.142 + 0.896i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.862 - 0.506i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 0.862 - 0.506i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.178608418\)
\(L(\frac12)\) \(\approx\) \(2.178608418\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (0.951 - 0.309i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.587 - 0.809i)T^{2} \)
13 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.587 - 0.809i)T^{2} \)
29 \( 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2} \)
31 \( 1 + (-0.951 - 0.309i)T^{2} \)
37 \( 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2} \)
41 \( 1 + (1.76 + 0.896i)T + (0.587 + 0.809i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.951 - 0.309i)T^{2} \)
73 \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.951 + 0.309i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2} \)
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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

−9.343007597449553080060098437759, −8.839573348371055477353546059787, −7.975160184183540512703303826268, −6.92146810719295920132731601460, −6.29311748627822302911422693490, −5.37655631530592154541727317910, −4.90386069319595101373230451012, −3.83979389728859768296722342993, −2.68568662589222422060577577307, −1.81012166213159151054660497339, 1.54229158408636676130282228954, 2.80214802724370521515956217042, 3.26181198649213658708302791762, 4.49984570788973186341281083380, 5.55600479057937034571006726346, 6.08708668275975580834125233031, 6.64253660407338528837044053857, 7.83368530234974541875017519008, 8.650708821574018866282941529601, 9.843471179915113507844852175649

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