Properties

Label 2-5070-13.12-c1-0-96
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 − 4i·7-s + i·8-s + 9-s − 10-s − 12-s − 4·14-s i·15-s + 16-s + 2·17-s i·18-s + 4i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s − 1.51i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.288·12-s − 1.06·14-s − 0.258i·15-s + 0.250·16-s + 0.485·17-s − 0.235i·18-s + 0.917i·19-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

\(L(1)\) \(\approx\) \(1.235259041\)
\(L(\frac12)\) \(\approx\) \(1.235259041\)
\(L(\frac{3}{2})\) 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 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 8iT - 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 16iT - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 14iT - 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−7.902779101384107378741333793771, −7.38974578229457086264478775725, −6.36681463879856556672251198774, −5.50535974498790619140837492969, −4.46948761712770069045514210303, −3.93742026584893608206283595605, −3.41461401347423865146825602999, −2.16965592687440240716886965186, −1.37644351106380768032526982252, −0.29520426968162943152674472081, 1.61429504050987896324182304044, 2.64268412581824663025404423360, 3.23796013966328258236053777514, 4.31199544402349728218557219639, 5.16922959109749358835537838779, 5.81744961166176221543510098640, 6.55477693633447694789149361466, 7.16404990992205796268208567653, 8.156534110156454528291463361006, 8.454568999041999285548090878256

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