Properties

Label 2-5070-13.12-c1-0-21
Degree $2$
Conductor $5070$
Sign $-0.277 - 0.960i$
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 + 0.464i·11-s + 12-s + 2·14-s i·15-s + 16-s − 4·17-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 + 0.139i·11-s + 0.288·12-s + 0.534·14-s − 0.258i·15-s + 0.250·16-s − 0.970·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.277 - 0.960i$
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: \((2,\ 5070,\ (\ :1/2),\ -0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.054056787\)
\(L(\frac12)\) \(\approx\) \(1.054056787\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.464iT - 11T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 0.535iT - 19T^{2} \)
23 \( 1 - 0.267T + 23T^{2} \)
29 \( 1 - 3.73T + 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 - 1.19iT - 37T^{2} \)
41 \( 1 + 2iT - 41T^{2} \)
43 \( 1 + 1.92T + 43T^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 - 1.53iT - 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 4.53iT - 67T^{2} \)
71 \( 1 - 8.39iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 0.0717T + 79T^{2} \)
83 \( 1 + 4.92iT - 83T^{2} \)
89 \( 1 - 7.46iT - 89T^{2} \)
97 \( 1 - 7.46iT - 97T^{2} \)
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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

−8.335086245811485636338093679666, −7.50354576580451273373249230230, −6.87853040177960279467284463646, −6.50062866834250144056278793920, −5.62500485337379156825898025440, −4.83170884609190271036764818224, −4.17672741057560262241094813117, −3.35434524088864004605204738591, −2.10442867404915427308840673302, −0.799226027606984706254904247300, 0.42419887648281060949837645476, 1.60756357944892055932219947748, 2.47814259297317977755325093201, 3.40559991718207886799801950101, 4.51017453761159592094196942276, 4.85923643996565212245001478707, 5.83955675128917174586473787014, 6.33686905891699146741406733039, 7.33510924233946762497410306890, 8.249316013278847959145133291050

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