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Properties

Label 2-5070-13.12-c1-0-9
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 + 3i·7-s − i·8-s + 9-s − 10-s − i·11-s + 12-s − 3·14-s − i·15-s + 16-s + i·18-s − 5i·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 + 1.13i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.301i·11-s + 0.288·12-s − 0.801·14-s − 0.258i·15-s + 0.250·16-s + 0.235i·18-s − 1.14i·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

\(L(1)\) \(\approx\) \(0.7234082448\)
\(L(\frac12)\) \(\approx\) \(0.7234082448\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
37 \( 1 - iT - 37T^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 9iT - 47T^{2} \)
53 \( 1 + 13T + 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + iT - 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−8.624455572400189252113725503015, −7.890366130486890043551756101203, −6.99872038361984360056429601323, −6.51242010608576055860759163538, −5.85568131055190993830714123800, −5.09009880763016517780845986568, −4.61578999069602632734149920491, −3.31062367643283639200029230308, −2.64017374191209443633296919649, −1.26677341228616871085555583614, 0.24066229365829699213152901968, 1.18608190003359032364789492095, 2.08655138042056272995658128936, 3.38394702273183657784340783046, 4.10920311006230030776778162191, 4.64830485734619457871118172859, 5.54170786465685025551120874055, 6.22694060213884888219550318361, 7.28116974565876885695440782314, 7.67582679953680771177977052102

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