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Properties

Label 2-700-140.139-c1-0-47
Degree $2$
Conductor $700$
Sign $-0.608 + 0.793i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.366i)2-s + 1.73i·3-s + (1.73 − i)4-s + (−0.633 − 2.36i)6-s + (−2 − 1.73i)7-s + (−1.99 + 2i)8-s + 0.267i·11-s + (1.73 + 2.99i)12-s + 0.464·13-s + (3.36 + 1.63i)14-s + (1.99 − 3.46i)16-s − 6.46·17-s − 6·19-s + (2.99 − 3.46i)21-s + (−0.0980 − 0.366i)22-s + 1.46·23-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + 0.999i·3-s + (0.866 − 0.5i)4-s + (−0.258 − 0.965i)6-s + (−0.755 − 0.654i)7-s + (−0.707 + 0.707i)8-s + 0.0807i·11-s + (0.499 + 0.866i)12-s + 0.128·13-s + (0.899 + 0.436i)14-s + (0.499 − 0.866i)16-s − 1.56·17-s − 1.37·19-s + (0.654 − 0.755i)21-s + (−0.0209 − 0.0780i)22-s + 0.305·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.608 + 0.793i$
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: \(1\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
5 \( 1 \)
7 \( 1 + (2 + 1.73i)T \)
good3 \( 1 - 1.73iT - 3T^{2} \)
11 \( 1 - 0.267iT - 11T^{2} \)
13 \( 1 - 0.464T + 13T^{2} \)
17 \( 1 + 6.46T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 9.46iT - 37T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 1.73iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 + 9.46iT - 61T^{2} \)
67 \( 1 - 3.46T + 67T^{2} \)
71 \( 1 - 7.46iT - 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 + 15.4iT - 83T^{2} \)
89 \( 1 - 2.53iT - 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−10.05799512694832304313605799960, −9.252104418130801090492885296038, −8.815423881410872495270363876489, −7.47956817201514340032163902001, −6.78787963755710182953511468169, −5.84047545893985975467415757098, −4.52267099842250700225821745564, −3.60443275592825119503335231720, −2.06832379625223456238426573275, 0, 1.76987501835780290750468231858, 2.60952557767775544407385728402, 4.02684444136377188015058673705, 5.84482706618288067489072645249, 6.68459994049540358915687053246, 7.16874214232023280176066163320, 8.351594504223908989872080544213, 8.911527373687047653955969768217, 9.765414481283198575973895499354

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