Properties

Label 2-5070-13.12-c1-0-17
Degree $2$
Conductor $5070$
Sign $0.999 + 0.0304i$
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 − 1.69i·7-s + i·8-s + 9-s − 10-s + 2.15i·11-s + 12-s − 1.69·14-s + i·15-s + 16-s − 2.35·17-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 − 0.639i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.650i·11-s + 0.288·12-s − 0.452·14-s + 0.258i·15-s + 0.250·16-s − 0.571·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5070\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $0.999 + 0.0304i$
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: \((2,\ 5070,\ (\ :1/2),\ 0.999 + 0.0304i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.007077148\)
\(L(\frac12)\) \(\approx\) \(1.007077148\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.69iT - 7T^{2} \)
11 \( 1 - 2.15iT - 11T^{2} \)
17 \( 1 + 2.35T + 17T^{2} \)
19 \( 1 + 0.198iT - 19T^{2} \)
23 \( 1 - 3.74T + 23T^{2} \)
29 \( 1 - 1.29T + 29T^{2} \)
31 \( 1 + 1.44iT - 31T^{2} \)
37 \( 1 - 0.801iT - 37T^{2} \)
41 \( 1 - 1.89iT - 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 - 8.87iT - 47T^{2} \)
53 \( 1 + 1.00T + 53T^{2} \)
59 \( 1 - 3.73iT - 59T^{2} \)
61 \( 1 + 6.32T + 61T^{2} \)
67 \( 1 - 7.56iT - 67T^{2} \)
71 \( 1 - 4.18iT - 71T^{2} \)
73 \( 1 - 11.9iT - 73T^{2} \)
79 \( 1 - 9.40T + 79T^{2} \)
83 \( 1 + 8.43iT - 83T^{2} \)
89 \( 1 + 2.41iT - 89T^{2} \)
97 \( 1 - 0.0881iT - 97T^{2} \)
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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

−8.328583871964921893755762157264, −7.46609189306214587646881017004, −6.82918955875806403263267639958, −5.99569165707473943228000165010, −5.01830314155687045643724091887, −4.57431409359409685413236435977, −3.83833589739774039399396316305, −2.80257865950877720234005857678, −1.71894822255597981223504137115, −0.854798684823446662482882527467, 0.38339072863749065379971307672, 1.82666995166732522605061441393, 2.99443411268074903542806114506, 3.79689068299856009326087889439, 4.90013502628408115834863376116, 5.33205840067449704884021827834, 6.25491304195658427451508683299, 6.59930065085825128862335316072, 7.38861783402956083639687909114, 8.205989492928119290155535959835

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