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Properties

Label 2-700-140.139-c1-0-42
Degree $2$
Conductor $700$
Sign $0.320 + 0.947i$
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  + (−0.736 + 1.20i)2-s − 2.79i·3-s + (−0.914 − 1.77i)4-s + (3.37 + 2.06i)6-s + (2.51 − 0.819i)7-s + (2.82 + 0.207i)8-s − 4.82·9-s − 1.47i·11-s + (−4.97 + 2.55i)12-s + 5.83·13-s + (−0.864 + 3.64i)14-s + (−2.32 + 3.25i)16-s + 4.12·17-s + (3.55 − 5.82i)18-s + 5.11·19-s + ⋯
L(s)  = 1  + (−0.521 + 0.853i)2-s − 1.61i·3-s + (−0.457 − 0.889i)4-s + (1.37 + 0.841i)6-s + (0.950 − 0.309i)7-s + (0.997 + 0.0732i)8-s − 1.60·9-s − 0.444i·11-s + (−1.43 + 0.738i)12-s + 1.61·13-s + (−0.231 + 0.972i)14-s + (−0.582 + 0.813i)16-s + 0.999·17-s + (0.838 − 1.37i)18-s + 1.17·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.320 + 0.947i$
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: \((2,\ 700,\ (\ :1/2),\ 0.320 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04677 - 0.751091i\)
\(L(\frac12)\) \(\approx\) \(1.04677 - 0.751091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.736 - 1.20i)T \)
5 \( 1 \)
7 \( 1 + (-2.51 + 0.819i)T \)
good3 \( 1 + 2.79iT - 3T^{2} \)
11 \( 1 + 1.47iT - 11T^{2} \)
13 \( 1 - 5.83T + 13T^{2} \)
17 \( 1 - 4.12T + 17T^{2} \)
19 \( 1 - 5.11T + 19T^{2} \)
23 \( 1 + 2.08T + 23T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 + 3.95T + 31T^{2} \)
37 \( 1 - 2.24iT - 37T^{2} \)
41 \( 1 + 4.12iT - 41T^{2} \)
43 \( 1 + 2.94T + 43T^{2} \)
47 \( 1 + 11.8iT - 47T^{2} \)
53 \( 1 - 3.75iT - 53T^{2} \)
59 \( 1 + 5.59T + 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 - 7.97iT - 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 - 4.16iT - 79T^{2} \)
83 \( 1 + 2.79iT - 83T^{2} \)
89 \( 1 + 12.3iT - 89T^{2} \)
97 \( 1 - 8.24T + 97T^{2} \)
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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

−10.25509907323499581830871205724, −8.940261879692425234132753594302, −8.283401593272986739177591028615, −7.60262648385252165511336683000, −7.04367188979692652657914580417, −5.89970685293003003705017860028, −5.44906548418690981838198331543, −3.70472098979283843068489879335, −1.73477651051829381818562220744, −0.958816106217446426112407318102, 1.57716017248780894564658079409, 3.23978124767372462922344847760, 3.91081311977405158111473116679, 4.90789815484433520298494991396, 5.73968961837536451054422991751, 7.61833338598916072731452359367, 8.331933311312365573779347019186, 9.297653560490184784037495190886, 9.662816443661228276820994914691, 10.73069245369064335557937095617

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