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Properties

Label 2-700-140.103-c0-0-1
Degree $2$
Conductor $700$
Sign $-0.350 + 0.936i$
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  + (−0.965 + 0.258i)2-s + (0.448 − 1.67i)3-s + (0.866 − 0.499i)4-s + 1.73i·6-s + (0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + (−1.73 − 1.00i)9-s + (−0.448 − 1.67i)12-s + i·14-s + (0.500 − 0.866i)16-s + (1.93 + 0.517i)18-s + (−1.50 − 0.866i)21-s + (0.258 + 0.965i)23-s + (0.866 + 1.5i)24-s + (−1.22 + 1.22i)27-s + (−0.258 − 0.965i)28-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.448 − 1.67i)3-s + (0.866 − 0.499i)4-s + 1.73i·6-s + (0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + (−1.73 − 1.00i)9-s + (−0.448 − 1.67i)12-s + i·14-s + (0.500 − 0.866i)16-s + (1.93 + 0.517i)18-s + (−1.50 − 0.866i)21-s + (0.258 + 0.965i)23-s + (0.866 + 1.5i)24-s + (−1.22 + 1.22i)27-s + (−0.258 − 0.965i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.350 + 0.936i$
Analytic conductor: \(0.349345\)
Root analytic conductor: \(0.591054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :0),\ -0.350 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7054093129\)
\(L(\frac12)\) \(\approx\) \(0.7054093129\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-0.258 + 0.965i)T \)
good3 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
89 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.36268789689117360710872113655, −9.265864009517202178427950550137, −8.443862449739234824550584669835, −7.68741034034441153865189504168, −7.13770622932112209241381618560, −6.49582564073515510279271548594, −5.34369959802133308087032788632, −3.39575440047677030332807975179, −2.06628930928276043501562760265, −1.06083887717798440944089107709, 2.30808971342957337638530372338, 3.21083210508558674995845129852, 4.35560229737969448247255841469, 5.41747033425008780839255495450, 6.53301242994913291821968430282, 8.063965335894345063377208482395, 8.526729682499672177085527438316, 9.362931968819186807178214055089, 9.869761660819311228290279299605, 10.69619795725360759205812252809

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