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Properties

Label 2-700-20.3-c1-0-53
Degree $2$
Conductor $700$
Sign $-0.536 + 0.843i$
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  + (1.15 − 0.819i)2-s + (1.96 − 1.96i)3-s + (0.656 − 1.88i)4-s + (0.653 − 3.87i)6-s + (−0.707 − 0.707i)7-s + (−0.792 − 2.71i)8-s − 4.71i·9-s + 0.965i·11-s + (−2.42 − 5.00i)12-s + (3.26 + 3.26i)13-s + (−1.39 − 0.235i)14-s + (−3.13 − 2.48i)16-s + (−5.11 + 5.11i)17-s + (−3.86 − 5.43i)18-s + 5.57·19-s + ⋯
L(s)  = 1  + (0.814 − 0.579i)2-s + (1.13 − 1.13i)3-s + (0.328 − 0.944i)4-s + (0.266 − 1.58i)6-s + (−0.267 − 0.267i)7-s + (−0.280 − 0.959i)8-s − 1.57i·9-s + 0.291i·11-s + (−0.699 − 1.44i)12-s + (0.904 + 0.904i)13-s + (−0.372 − 0.0628i)14-s + (−0.784 − 0.620i)16-s + (−1.24 + 1.24i)17-s + (−0.911 − 1.28i)18-s + 1.27·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.536 + 0.843i$
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: \((2,\ 700,\ (\ :1/2),\ -0.536 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58964 - 2.89400i\)
\(L(\frac12)\) \(\approx\) \(1.58964 - 2.89400i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 + 0.819i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-1.96 + 1.96i)T - 3iT^{2} \)
11 \( 1 - 0.965iT - 11T^{2} \)
13 \( 1 + (-3.26 - 3.26i)T + 13iT^{2} \)
17 \( 1 + (5.11 - 5.11i)T - 17iT^{2} \)
19 \( 1 - 5.57T + 19T^{2} \)
23 \( 1 + (2.07 - 2.07i)T - 23iT^{2} \)
29 \( 1 - 3.83iT - 29T^{2} \)
31 \( 1 + 6.27iT - 31T^{2} \)
37 \( 1 + (4.43 - 4.43i)T - 37iT^{2} \)
41 \( 1 - 2.49T + 41T^{2} \)
43 \( 1 + (-8.83 + 8.83i)T - 43iT^{2} \)
47 \( 1 + (5.81 + 5.81i)T + 47iT^{2} \)
53 \( 1 + (-1.36 - 1.36i)T + 53iT^{2} \)
59 \( 1 - 6.59T + 59T^{2} \)
61 \( 1 + 2.61T + 61T^{2} \)
67 \( 1 + (-2.43 - 2.43i)T + 67iT^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (7.01 + 7.01i)T + 73iT^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + (4.01 - 4.01i)T - 83iT^{2} \)
89 \( 1 + 12.0iT - 89T^{2} \)
97 \( 1 + (2.77 - 2.77i)T - 97iT^{2} \)
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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

−10.20496729704346855087744878614, −9.209086420024877045955086331437, −8.523167529819069853924673130116, −7.30844128200553718252967411424, −6.70535925103366640167461098986, −5.77631470985185966170148569871, −4.19579244725692858285770089019, −3.45546096862930186860869774758, −2.23592473825490274686534593424, −1.39973574486541078316575739339, 2.65843937943284904810345633446, 3.26465454355786061993904007723, 4.25102060668804404861455534798, 5.13581459439596599106817924435, 6.12513009641250485341428170340, 7.33474790129545656397443859530, 8.241902170708329580456053539822, 8.916029323594479765253816381657, 9.638158500815146331441330206093, 10.76414630043084853734610582245

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