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Properties

Label 2-2960-740.219-c0-0-0
Degree $2$
Conductor $2960$
Sign $-0.238 - 0.971i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
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  + (0.5 + 0.866i)5-s + (−0.173 + 0.984i)9-s + (−1.70 + 0.300i)13-s + (0.673 + 0.118i)17-s + (−0.499 + 0.866i)25-s + (−0.173 + 0.300i)29-s + (0.173 + 0.984i)37-s + (0.326 + 1.85i)41-s + (−0.939 + 0.342i)45-s + (−0.766 − 0.642i)49-s + (−0.592 − 1.62i)53-s + (−0.0603 − 0.342i)61-s + (−1.11 − 1.32i)65-s + 1.73i·73-s + (−0.939 − 0.342i)81-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (−0.173 + 0.984i)9-s + (−1.70 + 0.300i)13-s + (0.673 + 0.118i)17-s + (−0.499 + 0.866i)25-s + (−0.173 + 0.300i)29-s + (0.173 + 0.984i)37-s + (0.326 + 1.85i)41-s + (−0.939 + 0.342i)45-s + (−0.766 − 0.642i)49-s + (−0.592 − 1.62i)53-s + (−0.0603 − 0.342i)61-s + (−1.11 − 1.32i)65-s + 1.73i·73-s + (−0.939 − 0.342i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.238 - 0.971i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ -0.238 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.073762912\)
\(L(\frac12)\) \(\approx\) \(1.073762912\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.173 - 0.984i)T \)
good3 \( 1 + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.766 + 0.642i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
19 \( 1 + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.939 - 0.342i)T^{2} \)
89 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (-1.70 + 0.984i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.342970442838997783641519704969, −8.189318631846809404344429868787, −7.63450614276196363248057008342, −6.92289170968070159009751576550, −6.18013312781564511632797836558, −5.19740456099567340326347014194, −4.71392421655945255805320476196, −3.36006806512681040598811595529, −2.58389485632583811498182655909, −1.78280011909008805058367522047, 0.63792047485703077912084748771, 1.99479666487297060881955332557, 2.98023392957289680448694892082, 4.06177365460030625128401030788, 4.91699345180268470383801249049, 5.61350730012024056045731543384, 6.29592894752241670018291373551, 7.36612355848177678278158711701, 7.85504520248727368614404572025, 9.024803872625387357777787651301

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