Properties

Label 2-5070-13.12-c1-0-35
Degree $2$
Conductor $5070$
Sign $0.832 - 0.554i$
Analytic cond. $40.4841$
Root an. cond. $6.36271$
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  + i·2-s − 3-s − 4-s i·5-s i·6-s − 2i·7-s i·8-s + 9-s + 10-s i·11-s + 12-s + 2·14-s + i·15-s + 16-s − 2·17-s + i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s − 0.755i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s − 0.301i·11-s + 0.288·12-s + 0.534·14-s + 0.258i·15-s + 0.250·16-s − 0.485·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $0.832 - 0.554i$
Analytic conductor: \(40.4841\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5070} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5070,\ (\ :1/2),\ 0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.254697384\)
\(L(\frac12)\) \(\approx\) \(1.254697384\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + T \)
5 \( 1 + iT \)
13 \( 1 \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + iT - 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 3iT - 31T^{2} \)
37 \( 1 - 5iT - 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 14T + 53T^{2} \)
59 \( 1 - 5iT - 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 4iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 10iT - 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−8.233431023005008434394608399032, −7.47771080534668747785694431991, −6.87456564460556029096669783655, −6.07817884334983330702626992726, −5.54895163414580302947710408022, −4.64005238413911631868106589778, −4.12672951268735207937340952666, −3.18601641569414944243713808471, −1.65360362690956341577402281819, −0.67003337186455975986198714174, 0.60179568925753632161282610233, 1.97122151110831402224764351116, 2.61956663383141666578452604900, 3.56059420124176434576397231008, 4.49918098312828972745521688840, 5.19341971990176875015362820515, 5.84042294540005821525179389638, 6.82400945157533984914835099993, 7.23737362341621947535403576022, 8.349473859479836628470175526276

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