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Properties

Label 2-700-5.2-c2-0-12
Degree $2$
Conductor $700$
Sign $0.229 + 0.973i$
Analytic cond. $19.0736$
Root an. cond. $4.36733$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.87 + 2.87i)3-s + (1.87 + 1.87i)7-s − 7.48i·9-s − 3.51·11-s + (−0.612 + 0.612i)13-s + (−14.0 − 14.0i)17-s + 2.25i·19-s − 10.7·21-s + (−29.7 + 29.7i)23-s + (−4.35 − 4.35i)27-s − 16.9i·29-s + 33.6·31-s + (10.0 − 10.0i)33-s + (−14.7 − 14.7i)37-s − 3.51i·39-s + ⋯
L(s)  = 1  + (−0.956 + 0.956i)3-s + (0.267 + 0.267i)7-s − 0.831i·9-s − 0.319·11-s + (−0.0471 + 0.0471i)13-s + (−0.829 − 0.829i)17-s + 0.118i·19-s − 0.511·21-s + (−1.29 + 1.29i)23-s + (−0.161 − 0.161i)27-s − 0.583i·29-s + 1.08·31-s + (0.305 − 0.305i)33-s + (−0.399 − 0.399i)37-s − 0.0901i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(19.0736\)
Root analytic conductor: \(4.36733\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4126684581\)
\(L(\frac12)\) \(\approx\) \(0.4126684581\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (-1.87 - 1.87i)T \)
good3 \( 1 + (2.87 - 2.87i)T - 9iT^{2} \)
11 \( 1 + 3.51T + 121T^{2} \)
13 \( 1 + (0.612 - 0.612i)T - 169iT^{2} \)
17 \( 1 + (14.0 + 14.0i)T + 289iT^{2} \)
19 \( 1 - 2.25iT - 361T^{2} \)
23 \( 1 + (29.7 - 29.7i)T - 529iT^{2} \)
29 \( 1 + 16.9iT - 841T^{2} \)
31 \( 1 - 33.6T + 961T^{2} \)
37 \( 1 + (14.7 + 14.7i)T + 1.36e3iT^{2} \)
41 \( 1 + 5.67T + 1.68e3T^{2} \)
43 \( 1 + (-38.4 + 38.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (49.7 + 49.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-53.7 + 53.7i)T - 2.80e3iT^{2} \)
59 \( 1 - 39.2iT - 3.48e3T^{2} \)
61 \( 1 + 98.9T + 3.72e3T^{2} \)
67 \( 1 + (-31.2 - 31.2i)T + 4.48e3iT^{2} \)
71 \( 1 - 138.T + 5.04e3T^{2} \)
73 \( 1 + (-79.4 + 79.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 151. iT - 6.24e3T^{2} \)
83 \( 1 + (42.9 - 42.9i)T - 6.88e3iT^{2} \)
89 \( 1 + 117. iT - 7.92e3T^{2} \)
97 \( 1 + (42.3 + 42.3i)T + 9.40e3iT^{2} \)
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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

−10.09229957925814143117296417040, −9.491138601671185427954051645226, −8.424920002489977405200979928347, −7.41239276791746491630212527368, −6.24130501182097605604810641459, −5.42329848618631698928835763186, −4.69173697623638667940781308692, −3.72314519165070437506503815103, −2.17561805324288940297534856810, −0.18094683282675033443062977800, 1.16966620294801121838136904517, 2.42410729923963878812214649680, 4.11053084263596782838712946526, 5.11851300919088623136068091824, 6.26223968667620232887073390362, 6.63757075267476662144193361303, 7.79393778454083234684384858853, 8.423609239401519962260723749286, 9.712811576915664410880670327971, 10.74707713208799123437806250070

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