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Properties

Label 2-1960-1960.1339-c0-0-0
Degree $2$
Conductor $1960$
Sign $-0.814 + 0.580i$
Analytic cond. $0.978167$
Root an. cond. $0.989023$
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  + (0.365 + 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.826 − 0.563i)5-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (−0.988 + 0.149i)9-s + (0.826 + 0.563i)10-s + (−1.63 − 0.246i)11-s + (−0.914 + 1.14i)13-s + (−0.733 − 0.680i)14-s + (0.0747 − 0.997i)16-s + (−0.5 − 0.866i)18-s + (−0.0747 + 0.129i)19-s + (−0.222 + 0.974i)20-s + (−0.367 − 1.61i)22-s + (−0.955 − 0.294i)23-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.826 − 0.563i)5-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (−0.988 + 0.149i)9-s + (0.826 + 0.563i)10-s + (−1.63 − 0.246i)11-s + (−0.914 + 1.14i)13-s + (−0.733 − 0.680i)14-s + (0.0747 − 0.997i)16-s + (−0.5 − 0.866i)18-s + (−0.0747 + 0.129i)19-s + (−0.222 + 0.974i)20-s + (−0.367 − 1.61i)22-s + (−0.955 − 0.294i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.814 + 0.580i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (1339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ -0.814 + 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3522714076\)
\(L(\frac12)\) \(\approx\) \(0.3522714076\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.365 - 0.930i)T \)
5 \( 1 + (-0.826 + 0.563i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
good3 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (1.63 + 0.246i)T + (0.955 + 0.294i)T^{2} \)
13 \( 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.826 + 0.563i)T^{2} \)
19 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.955 + 0.294i)T + (0.826 + 0.563i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.44 - 1.34i)T + (0.0747 + 0.997i)T^{2} \)
41 \( 1 + (0.658 + 0.317i)T + (0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.698 - 1.77i)T + (-0.733 + 0.680i)T^{2} \)
53 \( 1 + (1.40 - 1.29i)T + (0.0747 - 0.997i)T^{2} \)
59 \( 1 + (1.48 + 1.01i)T + (0.365 + 0.930i)T^{2} \)
61 \( 1 + (-0.0747 - 0.997i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.733 + 0.680i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.440 + 0.0663i)T + (0.955 - 0.294i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.566113985582372989090520886958, −9.018496901257992578423344494640, −8.192637935659126261290864592726, −7.56783469862945081843887860296, −6.22542646764232867791134756597, −6.10716676432385820955039035119, −5.11023381141782802354097016801, −4.53657204789824760457614206764, −3.02565455427558259759484308449, −2.39469410897037642133894825810, 0.19902045935410455131523431626, 2.27715502030242889130615164195, 2.79339634448406453596953142870, 3.54427174513315786493887347316, 4.93920734955879224789332593976, 5.62368862819652321872358637885, 6.16914228302165848516186650516, 7.37862522084780706568428514302, 8.174270880499363721874432782858, 9.313342616635096706902960912516

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