Properties

Label 2-1400-1400.1021-c0-0-1
Degree $2$
Conductor $1400$
Sign $0.535 + 0.844i$
Analytic cond. $0.698691$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (−1.30 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (−1.30 + 0.951i)6-s + 7-s + (−0.809 + 0.587i)8-s + (0.500 + 1.53i)9-s + (0.809 + 0.587i)10-s + (0.499 + 1.53i)12-s + (0.5 + 1.53i)13-s + (0.309 − 0.951i)14-s + (1.30 − 0.951i)15-s + (0.309 + 0.951i)16-s + 1.61·18-s + (1.61 − 1.17i)19-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−1.30 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (−1.30 + 0.951i)6-s + 7-s + (−0.809 + 0.587i)8-s + (0.500 + 1.53i)9-s + (0.809 + 0.587i)10-s + (0.499 + 1.53i)12-s + (0.5 + 1.53i)13-s + (0.309 − 0.951i)14-s + (1.30 − 0.951i)15-s + (0.309 + 0.951i)16-s + 1.61·18-s + (1.61 − 1.17i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.535 + 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7769386574\)
\(L(\frac12)\) \(\approx\) \(0.7769386574\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 - T \)
good3 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−9.891721024535960274384115057480, −9.016026076657478319656677813038, −7.75126551507613395142754451636, −7.12171918416976001221394753408, −6.25137562299408477533209363567, −5.42355156779689226948832990600, −4.63179191282320981710034938650, −3.53212327050473597218943166149, −2.14507414811926742479397129886, −1.25102279053126631379284794267, 0.856756275800495936277887822617, 3.49296635226964067475845602398, 4.35538968792354698104366005420, 5.06861614393463777531848420907, 5.54470601327486007760926593254, 6.18801518191244997682381618804, 7.62885541190765653042439619714, 8.111857166367127056275971952473, 8.948686116605942740699993239210, 9.953256462044768592643011694813

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