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Properties

Label 2-1520-1520.949-c0-0-10
Degree $2$
Conductor $1520$
Sign $-0.923 + 0.382i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.831 − 0.555i)2-s + (0.785 − 0.785i)3-s + (0.382 + 0.923i)4-s + (−0.707 − 0.707i)5-s + (−1.08 + 0.216i)6-s + (0.195 − 0.980i)8-s − 0.234i·9-s + (0.195 + 0.980i)10-s + (−1.30 − 1.30i)11-s + (1.02 + 0.425i)12-s + (1.38 − 1.38i)13-s − 1.11·15-s + (−0.707 + 0.707i)16-s + (−0.130 + 0.195i)18-s + (−0.707 + 0.707i)19-s + (0.382 − 0.923i)20-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)2-s + (0.785 − 0.785i)3-s + (0.382 + 0.923i)4-s + (−0.707 − 0.707i)5-s + (−1.08 + 0.216i)6-s + (0.195 − 0.980i)8-s − 0.234i·9-s + (0.195 + 0.980i)10-s + (−1.30 − 1.30i)11-s + (1.02 + 0.425i)12-s + (1.38 − 1.38i)13-s − 1.11·15-s + (−0.707 + 0.707i)16-s + (−0.130 + 0.195i)18-s + (−0.707 + 0.707i)19-s + (0.382 − 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7231306567\)
\(L(\frac12)\) \(\approx\) \(0.7231306567\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.831 + 0.555i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
13 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.785 + 0.785i)T + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
67 \( 1 + (0.275 - 0.275i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 1.96T + T^{2} \)
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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

−8.853587714165728349214530844088, −8.412678870263727876719145416210, −8.050569945949492551594572272467, −7.47971454889551203596345524254, −6.20390487240215959467938309661, −5.18904379632705012795931440582, −3.60440909700402304676845354732, −3.18084988884894885121461940521, −1.93075877625395566105882929747, −0.68363604947588284655317275906, 1.98075904857857318105030422737, 3.02126186809074700372058332864, 4.18978102555595036663899147256, 4.86507697778398565923703431181, 6.33370909088916412787803625557, 6.90959861256840076627748756104, 7.75297221632741257973965380260, 8.533169853382861111460200759358, 9.042469848635564299145687971065, 9.981632005014164708582660580846

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