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Properties

Label 2-700-20.3-c1-0-4
Degree $2$
Conductor $700$
Sign $-0.827 - 0.561i$
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  + (−0.802 − 1.16i)2-s + (−1.75 + 1.75i)3-s + (−0.713 + 1.86i)4-s + (3.45 + 0.637i)6-s + (−0.707 − 0.707i)7-s + (2.74 − 0.667i)8-s − 3.17i·9-s + 5.34i·11-s + (−2.02 − 4.53i)12-s + (2.61 + 2.61i)13-s + (−0.256 + 1.39i)14-s + (−2.98 − 2.66i)16-s + (1.04 − 1.04i)17-s + (−3.69 + 2.54i)18-s + 0.260·19-s + ⋯
L(s)  = 1  + (−0.567 − 0.823i)2-s + (−1.01 + 1.01i)3-s + (−0.356 + 0.934i)4-s + (1.41 + 0.260i)6-s + (−0.267 − 0.267i)7-s + (0.971 − 0.235i)8-s − 1.05i·9-s + 1.61i·11-s + (−0.585 − 1.30i)12-s + (0.726 + 0.726i)13-s + (−0.0685 + 0.371i)14-s + (−0.745 − 0.666i)16-s + (0.252 − 0.252i)17-s + (−0.872 + 0.600i)18-s + 0.0597·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.827 - 0.561i$
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: \((2,\ 700,\ (\ :1/2),\ -0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105649 + 0.343576i\)
\(L(\frac12)\) \(\approx\) \(0.105649 + 0.343576i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.802 + 1.16i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.75 - 1.75i)T - 3iT^{2} \)
11 \( 1 - 5.34iT - 11T^{2} \)
13 \( 1 + (-2.61 - 2.61i)T + 13iT^{2} \)
17 \( 1 + (-1.04 + 1.04i)T - 17iT^{2} \)
19 \( 1 - 0.260T + 19T^{2} \)
23 \( 1 + (-1.06 + 1.06i)T - 23iT^{2} \)
29 \( 1 + 1.36iT - 29T^{2} \)
31 \( 1 + 2.05iT - 31T^{2} \)
37 \( 1 + (8.20 - 8.20i)T - 37iT^{2} \)
41 \( 1 + 9.70T + 41T^{2} \)
43 \( 1 + (6.86 - 6.86i)T - 43iT^{2} \)
47 \( 1 + (-0.976 - 0.976i)T + 47iT^{2} \)
53 \( 1 + (1.81 + 1.81i)T + 53iT^{2} \)
59 \( 1 - 4.53T + 59T^{2} \)
61 \( 1 + 4.13T + 61T^{2} \)
67 \( 1 + (4.96 + 4.96i)T + 67iT^{2} \)
71 \( 1 - 7.29iT - 71T^{2} \)
73 \( 1 + (9.65 + 9.65i)T + 73iT^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + (-5.83 + 5.83i)T - 83iT^{2} \)
89 \( 1 + 7.63iT - 89T^{2} \)
97 \( 1 + (11.5 - 11.5i)T - 97iT^{2} \)
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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

−10.59403675798560605890381913034, −10.04884195796518651743987193954, −9.538216675444461599151596772170, −8.530018809889733932677783828075, −7.30446584376181569975647179389, −6.43404473528315226689982789691, −4.94168925651201120956324044904, −4.41229145702500726215616075060, −3.35542373818280611244025424837, −1.68216083847090749071310767454, 0.27880788605716028661290752179, 1.45800452325142513043895193143, 3.45889753856665640783980700092, 5.38685429307123135914727619021, 5.71663577908787660581914307808, 6.55511236084694977514255208866, 7.29489199887699284858203956403, 8.372895995841082919860632708790, 8.840437630604245136984625721137, 10.24723829733415044956939429489

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