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Properties

Label 2-700-7.6-c0-0-1
Degree $2$
Conductor $700$
Sign $i$
Analytic cond. $0.349345$
Root an. cond. $0.591054$
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  − i·3-s − i·7-s − 11-s − i·13-s + i·17-s − 21-s − i·27-s + 29-s + i·33-s − 39-s + i·47-s − 49-s + 51-s + 2·71-s + 2i·73-s + ⋯
L(s)  = 1  − i·3-s − i·7-s − 11-s − i·13-s + i·17-s − 21-s − i·27-s + 29-s + i·33-s − 39-s + i·47-s − 49-s + 51-s + 2·71-s + 2i·73-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $i$
Analytic conductor: \(0.349345\)
Root analytic conductor: \(0.591054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9140492637\)
\(L(\frac12)\) \(\approx\) \(0.9140492637\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 + iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - 2iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−10.46066348006947616896795504313, −9.824947976513585554972070817681, −8.254657621720474302069963053615, −7.891889877853086117941671628195, −7.01449192328764480481317833813, −6.18675533791111212741284877628, −5.05183757213150740405063828488, −3.85203172566153858658551399481, −2.54749219910442087433699752843, −1.09331339495056724340748796325, 2.22972192861995293180341740539, 3.36215374059236355005664948042, 4.67109932167493845430517850771, 5.17433253965694296237475362312, 6.35368728470318625647203523095, 7.42766257220495881120390361797, 8.541282495342680240225583349039, 9.270719671494445086390622554280, 9.931782335995926821818685202912, 10.76035020396532464080243620846

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