Properties

Label 2-1520-5.4-c1-0-1
Degree $2$
Conductor $1520$
Sign $-0.929 + 0.369i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
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  + 0.537i·3-s + (−2.07 + 0.826i)5-s + 3.18i·7-s + 2.71·9-s − 4.15·11-s − 2.07i·13-s + (−0.443 − 1.11i)15-s + 5.79i·17-s − 19-s − 1.71·21-s − 2.60i·23-s + (3.63 − 3.43i)25-s + 3.06i·27-s − 6·29-s − 2.59·31-s + ⋯
L(s)  = 1  + 0.310i·3-s + (−0.929 + 0.369i)5-s + 1.20i·7-s + 0.903·9-s − 1.25·11-s − 0.574i·13-s + (−0.114 − 0.288i)15-s + 1.40i·17-s − 0.229·19-s − 0.373·21-s − 0.543i·23-s + (0.726 − 0.686i)25-s + 0.590i·27-s − 1.11·29-s − 0.466·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.929 + 0.369i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ -0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3842658039\)
\(L(\frac12)\) \(\approx\) \(0.3842658039\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.07 - 0.826i)T \)
19 \( 1 + T \)
good3 \( 1 - 0.537iT - 3T^{2} \)
7 \( 1 - 3.18iT - 7T^{2} \)
11 \( 1 + 4.15T + 11T^{2} \)
13 \( 1 + 2.07iT - 13T^{2} \)
17 \( 1 - 5.79iT - 17T^{2} \)
23 \( 1 + 2.60iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 2.59T + 31T^{2} \)
37 \( 1 - 4.30iT - 37T^{2} \)
41 \( 1 + 0.599T + 41T^{2} \)
43 \( 1 - 3.18iT - 43T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 + 1.71T + 59T^{2} \)
61 \( 1 + 8.75T + 61T^{2} \)
67 \( 1 + 4.76iT - 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 - 2.72iT - 73T^{2} \)
79 \( 1 + 1.40T + 79T^{2} \)
83 \( 1 + 7.07iT - 83T^{2} \)
89 \( 1 + 16.5T + 89T^{2} \)
97 \( 1 - 2.07iT - 97T^{2} \)
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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

−10.15710345438139353024610870546, −8.987911102440531551972259947322, −8.249025245421416970190821095416, −7.69859820250072258202066310837, −6.71972100826478792109785664968, −5.73056064158093050934180359985, −4.94228863464920909026787235515, −3.94507285876924600888578362166, −3.05706819607674734004801953736, −1.95983500701880417875286438867, 0.15303362611773562077013736034, 1.41654437170528093855821120845, 2.92064076815739047177013823143, 4.09952876691591596594841846598, 4.57243028722614835223907689665, 5.62993742511682734869543898016, 7.09589759313470583172300570115, 7.35153437686306517008449441922, 7.88294584280566883776204383468, 9.068972810720216837508058552250

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