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Properties

Label 2-5070-13.12-c1-0-46
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 + 3.56i·7-s + i·8-s + 9-s − 10-s − 4.12i·11-s − 12-s + 3.56·14-s − i·15-s + 16-s − 5.12·17-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 + 1.34i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 1.24i·11-s − 0.288·12-s + 0.951·14-s − 0.258i·15-s + 0.250·16-s − 1.24·17-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

\(L(1)\) \(\approx\) \(2.223611328\)
\(L(\frac12)\) \(\approx\) \(2.223611328\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.56iT - 7T^{2} \)
11 \( 1 + 4.12iT - 11T^{2} \)
17 \( 1 + 5.12T + 17T^{2} \)
19 \( 1 - 3.56iT - 19T^{2} \)
23 \( 1 - 7.68T + 23T^{2} \)
29 \( 1 - 6.56T + 29T^{2} \)
31 \( 1 + 5.68iT - 31T^{2} \)
37 \( 1 - 4.12iT - 37T^{2} \)
41 \( 1 - 4.24iT - 41T^{2} \)
43 \( 1 + 4.56T + 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 - 10.5iT - 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 14.2iT - 67T^{2} \)
71 \( 1 - 4.87iT - 71T^{2} \)
73 \( 1 + 15.3iT - 73T^{2} \)
79 \( 1 - 7.43T + 79T^{2} \)
83 \( 1 + 1.12iT - 83T^{2} \)
89 \( 1 - 1.80iT - 89T^{2} \)
97 \( 1 + 1.12iT - 97T^{2} \)
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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

−8.417564059307741006112365172192, −7.83197955891791611790452939328, −6.55205709379180738298393033216, −5.95357372469516743035332027857, −5.08236759008525371220487648706, −4.43629201704083080913939703847, −3.34833468681168197818866761913, −2.78246978519399196428664012748, −1.97571609449950513842209319654, −0.858645403548063982460919799732, 0.73970320132050704506355619621, 2.04087513681721084741147706522, 3.04394931967130554142983011255, 3.98431483478519103440254513412, 4.59034838786938362739342949573, 5.20768369016642425881875611213, 6.66576249717029833409861421799, 7.05869094008841827426696269902, 7.15126484992869238168008240004, 8.283233506015740282708905851421

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