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Properties

Label 2-700-28.3-c1-0-33
Degree $2$
Conductor $700$
Sign $0.890 + 0.455i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.836 − 1.14i)2-s + (−1.51 + 2.62i)3-s + (−0.601 + 1.90i)4-s + (4.25 − 0.465i)6-s + (2.57 − 0.602i)7-s + (2.67 − 0.908i)8-s + (−3.08 − 5.33i)9-s + (1.03 + 0.598i)11-s + (−4.08 − 4.46i)12-s − 4.83i·13-s + (−2.84 − 2.43i)14-s + (−3.27 − 2.29i)16-s + (−2.20 − 1.27i)17-s + (−3.51 + 7.97i)18-s + (−0.711 − 1.23i)19-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)2-s + (−0.873 + 1.51i)3-s + (−0.300 + 0.953i)4-s + (1.73 − 0.190i)6-s + (0.973 − 0.227i)7-s + (0.946 − 0.321i)8-s + (−1.02 − 1.77i)9-s + (0.312 + 0.180i)11-s + (−1.18 − 1.28i)12-s − 1.34i·13-s + (−0.759 − 0.650i)14-s + (−0.819 − 0.573i)16-s + (−0.534 − 0.308i)17-s + (−0.827 + 1.88i)18-s + (−0.163 − 0.282i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.890 + 0.455i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.792364 - 0.190766i\)
\(L(\frac12)\) \(\approx\) \(0.792364 - 0.190766i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.836 + 1.14i)T \)
5 \( 1 \)
7 \( 1 + (-2.57 + 0.602i)T \)
good3 \( 1 + (1.51 - 2.62i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.03 - 0.598i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.83iT - 13T^{2} \)
17 \( 1 + (2.20 + 1.27i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.711 + 1.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.02 + 2.90i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.774T + 29T^{2} \)
31 \( 1 + (-3.31 + 5.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.55 - 4.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.46iT - 41T^{2} \)
43 \( 1 - 1.38iT - 43T^{2} \)
47 \( 1 + (0.535 + 0.927i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.68 + 2.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.94 - 8.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.31 + 4.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.14 - 5.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 16.3iT - 71T^{2} \)
73 \( 1 + (0.0927 + 0.0535i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.32 + 5.38i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + (3.41 - 1.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.71iT - 97T^{2} \)
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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

−10.40210528331998702562922061893, −9.863279266647859488813551180964, −8.904847367369188795314248519680, −8.177271795593449647223613949268, −6.94101056625700683076648265758, −5.49391049114834537016936785505, −4.68273745736973960511899879060, −3.98460892553047401986061799485, −2.72468905740558807866741412921, −0.71528957953562981778827753616, 1.17778384430547960049771104980, 2.01260622243891480547895220090, 4.56406564564490900799819343384, 5.43334586465029872316603625734, 6.41162280951057939662421517434, 6.89028373267100292074836393485, 7.77808380768181141830520709699, 8.505130390455688697616747087020, 9.350509392177133406067427743510, 10.80914163934672621410838390612

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