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Properties

Label 2-700-28.27-c1-0-67
Degree $2$
Conductor $700$
Sign $-0.204 + 0.978i$
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  + (0.780 − 1.17i)2-s + 3.02·3-s + (−0.780 − 1.84i)4-s + (2.35 − 3.56i)6-s + (−2.17 − 1.51i)7-s + (−2.78 − 0.516i)8-s + 6.12·9-s − 4.71i·11-s + (−2.35 − 5.56i)12-s + 2i·13-s + (−3.47 + 1.38i)14-s + (−2.78 + 2.87i)16-s + 1.12i·17-s + (4.78 − 7.22i)18-s + 4.71·19-s + ⋯
L(s)  = 1  + (0.552 − 0.833i)2-s + 1.74·3-s + (−0.390 − 0.920i)4-s + (0.962 − 1.45i)6-s + (−0.821 − 0.570i)7-s + (−0.983 − 0.182i)8-s + 2.04·9-s − 1.42i·11-s + (−0.680 − 1.60i)12-s + 0.554i·13-s + (−0.929 + 0.369i)14-s + (−0.695 + 0.718i)16-s + 0.272i·17-s + (1.12 − 1.70i)18-s + 1.08·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95671 - 2.40894i\)
\(L(\frac12)\) \(\approx\) \(1.95671 - 2.40894i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.780 + 1.17i)T \)
5 \( 1 \)
7 \( 1 + (2.17 + 1.51i)T \)
good3 \( 1 - 3.02T + 3T^{2} \)
11 \( 1 + 4.71iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 1.12iT - 17T^{2} \)
19 \( 1 - 4.71T + 19T^{2} \)
23 \( 1 - 6.41iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 3.39T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 1.12iT - 41T^{2} \)
43 \( 1 + 0.371iT - 43T^{2} \)
47 \( 1 - 5.08T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 2.06T + 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 3.76iT - 67T^{2} \)
71 \( 1 - 7.36iT - 71T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 + 1.32iT - 79T^{2} \)
83 \( 1 - 3.02T + 83T^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 + 1.12iT - 97T^{2} \)
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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

−10.01005097973987661909179152982, −9.425028134667835682346492306547, −8.771191026426845775805579809183, −7.76898959288483181415114949872, −6.72000006691866918268031939104, −5.54341149780310330913117074260, −4.02956711218255853147007490599, −3.45697926533196343907930828810, −2.73898184045599622505822881641, −1.29392418476299362826843885936, 2.37914474750723851582928765253, 3.11691348429583146913087860206, 4.12428795407516981600653686686, 5.15868090834899321919807910996, 6.51435561024202441492314273616, 7.32013385011773901050774270842, 7.976253736948768882690439656190, 8.911758606649356703921069428009, 9.461827827051826118654686895329, 10.21755904491419071895435275410

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