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Properties

Label 2-700-140.139-c1-0-24
Degree $2$
Conductor $700$
Sign $0.615 + 0.788i$
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  + (−1.09 − 0.895i)2-s + (0.395 + 1.96i)4-s − 2.64·7-s + (1.32 − 2.49i)8-s + 3·9-s − 0.818i·11-s + (2.89 + 2.36i)14-s + (−3.68 + 1.55i)16-s + (−3.28 − 2.68i)18-s + (−0.732 + 0.895i)22-s + 4.28·23-s + (−1.04 − 5.18i)28-s + 8.16·29-s + (5.42 + 1.60i)32-s + (1.18 + 5.88i)36-s − 12.1i·37-s + ⋯
L(s)  = 1  + (−0.773 − 0.633i)2-s + (0.197 + 0.980i)4-s − 0.999·7-s + (0.467 − 0.883i)8-s + 9-s − 0.246i·11-s + (0.773 + 0.633i)14-s + (−0.921 + 0.387i)16-s + (−0.773 − 0.633i)18-s + (−0.156 + 0.190i)22-s + 0.892·23-s + (−0.197 − 0.980i)28-s + 1.51·29-s + (0.958 + 0.283i)32-s + (0.197 + 0.980i)36-s − 1.99i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.615 + 0.788i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.615 + 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881116 - 0.429990i\)
\(L(\frac12)\) \(\approx\) \(0.881116 - 0.429990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 + 0.895i)T \)
5 \( 1 \)
7 \( 1 + 2.64T \)
good3 \( 1 - 3T^{2} \)
11 \( 1 + 0.818iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4.28T + 23T^{2} \)
29 \( 1 - 8.16T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12.1iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 13.0T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 4.47T + 67T^{2} \)
71 \( 1 + 16.5iT - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 14.8iT - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.42298850098815579549283341818, −9.387537206100492383240954179120, −8.988059950397257764359539293953, −7.71251857102868754925926964089, −7.04356498512437000019837633107, −6.10921542296244409978738008425, −4.51013136184358079467491042034, −3.52605703607733055278328159876, −2.45423672349866192501160218437, −0.871912483507276550554667389136, 1.08741687263145026130161248014, 2.71257529886898354873588005543, 4.26030294860507599476217437847, 5.32817418583406912392422022002, 6.57363251531757928591524163258, 6.89996326746902680369313118144, 7.963446845920097104156664388806, 8.889815567098864159923321543368, 9.795732113070144719724336150268, 10.15243389012793390190420038157

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