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Properties

Label 2-700-20.3-c1-0-3
Degree $2$
Conductor $700$
Sign $-0.191 - 0.981i$
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  + (−0.297 − 1.38i)2-s + (−0.137 + 0.137i)3-s + (−1.82 + 0.823i)4-s + (0.231 + 0.149i)6-s + (−0.707 − 0.707i)7-s + (1.68 + 2.27i)8-s + 2.96i·9-s − 3.12i·11-s + (0.137 − 0.364i)12-s + (−2.50 − 2.50i)13-s + (−0.766 + 1.18i)14-s + (2.64 − 3.00i)16-s + (−2.85 + 2.85i)17-s + (4.09 − 0.882i)18-s − 3.29·19-s + ⋯
L(s)  = 1  + (−0.210 − 0.977i)2-s + (−0.0796 + 0.0796i)3-s + (−0.911 + 0.411i)4-s + (0.0945 + 0.0610i)6-s + (−0.267 − 0.267i)7-s + (0.594 + 0.804i)8-s + 0.987i·9-s − 0.942i·11-s + (0.0397 − 0.105i)12-s + (−0.694 − 0.694i)13-s + (−0.204 + 0.317i)14-s + (0.660 − 0.750i)16-s + (−0.693 + 0.693i)17-s + (0.965 − 0.207i)18-s − 0.755·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.191 - 0.981i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.191 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128246 + 0.155663i\)
\(L(\frac12)\) \(\approx\) \(0.128246 + 0.155663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.297 + 1.38i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.137 - 0.137i)T - 3iT^{2} \)
11 \( 1 + 3.12iT - 11T^{2} \)
13 \( 1 + (2.50 + 2.50i)T + 13iT^{2} \)
17 \( 1 + (2.85 - 2.85i)T - 17iT^{2} \)
19 \( 1 + 3.29T + 19T^{2} \)
23 \( 1 + (6.43 - 6.43i)T - 23iT^{2} \)
29 \( 1 + 3.55iT - 29T^{2} \)
31 \( 1 - 8.67iT - 31T^{2} \)
37 \( 1 + (3.64 - 3.64i)T - 37iT^{2} \)
41 \( 1 - 1.66T + 41T^{2} \)
43 \( 1 + (-1.17 + 1.17i)T - 43iT^{2} \)
47 \( 1 + (-0.598 - 0.598i)T + 47iT^{2} \)
53 \( 1 + (-1.12 - 1.12i)T + 53iT^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 6.81T + 61T^{2} \)
67 \( 1 + (-6.83 - 6.83i)T + 67iT^{2} \)
71 \( 1 - 3.15iT - 71T^{2} \)
73 \( 1 + (6.00 + 6.00i)T + 73iT^{2} \)
79 \( 1 + 0.153T + 79T^{2} \)
83 \( 1 + (-8.57 + 8.57i)T - 83iT^{2} \)
89 \( 1 + 17.4iT - 89T^{2} \)
97 \( 1 + (2.28 - 2.28i)T - 97iT^{2} \)
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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

−10.50607020274241731501474686139, −10.23474385910279895469224330336, −9.056714085539633593665102723229, −8.226799077022417860589054754267, −7.55237698422044896311797469752, −6.06693547522080470789369859395, −5.06721877144660068164478788818, −4.04032601116703296226032841833, −2.98227002988507352265209182267, −1.77952497015373999132686593574, 0.10993359492422404691382573216, 2.20195126497136089026828165116, 4.03374022583353070551438226838, 4.73111644081687969024836139075, 6.03571522525454817567534149491, 6.67231656834171013154394282269, 7.39838483537719960067472229839, 8.494243799188235445677293638914, 9.370431585094182720388303293850, 9.766849676002690379487905314763

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