Properties

Label 2-5070-13.12-c1-0-75
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 − 3i·7-s i·8-s + 9-s + 10-s + 0.267i·11-s + 12-s + 3·14-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 − 1.13i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 0.0807i·11-s + 0.288·12-s + 0.801·14-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

\(L(1)\) \(\approx\) \(1.288406290\)
\(L(\frac12)\) \(\approx\) \(1.288406290\)
\(L(\frac{3}{2})\) 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 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 0.267iT - 11T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 5.73iT - 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 1.46T + 29T^{2} \)
31 \( 1 + 4.92iT - 31T^{2} \)
37 \( 1 + 5.92iT - 37T^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 - 6.46iT - 47T^{2} \)
53 \( 1 + 0.267T + 53T^{2} \)
59 \( 1 - 11.4iT - 59T^{2} \)
61 \( 1 + 0.535T + 61T^{2} \)
67 \( 1 - 1.46iT - 67T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 3.07T + 79T^{2} \)
83 \( 1 + 9.46iT - 83T^{2} \)
89 \( 1 + 14.1iT - 89T^{2} \)
97 \( 1 + 8.39iT - 97T^{2} \)
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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

−7.77139306143961629822732876557, −7.34830887500170139093248588371, −6.77210902701531250532892507770, −5.84941242754529589314535899196, −5.28072999593660479153679792147, −4.42372900458056467662082465513, −3.97506642576963185699042054390, −2.76049507478416843875073529866, −1.19991264600636679015586514802, −0.46698173531477592581671371476, 1.13222105329963699464190322757, 2.08950087139785942106762745508, 3.05661034651306125954233263108, 3.66361966625167060608170690836, 4.82625416314967211969721486630, 5.41706139362196146515190430013, 6.08214175140015938558116170966, 6.79352433837296244084382057658, 7.85267949804141619713824758617, 8.356034466746222802947854315302

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