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Properties

Label 2-2960-740.587-c0-0-0
Degree $2$
Conductor $2960$
Sign $0.994 - 0.100i$
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  + (0.173 − 0.984i)5-s + (−0.642 + 0.766i)9-s + (0.642 + 0.766i)13-s + (1.50 + 1.26i)17-s + (−0.939 − 0.342i)25-s + (0.0451 + 0.168i)29-s + (0.766 − 0.642i)37-s + (1.26 + 1.50i)41-s + (0.642 + 0.766i)45-s + (−0.342 − 0.939i)49-s + (−1.10 − 1.58i)53-s + (1.98 + 0.173i)61-s + (0.866 − 0.5i)65-s + (−0.366 − 0.366i)73-s + (−0.173 − 0.984i)81-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)5-s + (−0.642 + 0.766i)9-s + (0.642 + 0.766i)13-s + (1.50 + 1.26i)17-s + (−0.939 − 0.342i)25-s + (0.0451 + 0.168i)29-s + (0.766 − 0.642i)37-s + (1.26 + 1.50i)41-s + (0.642 + 0.766i)45-s + (−0.342 − 0.939i)49-s + (−1.10 − 1.58i)53-s + (1.98 + 0.173i)61-s + (0.866 − 0.5i)65-s + (−0.366 − 0.366i)73-s + (−0.173 − 0.984i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.994 - 0.100i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.294630369\)
\(L(\frac12)\) \(\approx\) \(1.294630369\)
\(L(1)\) 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 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + (-0.766 + 0.642i)T \)
good3 \( 1 + (0.642 - 0.766i)T^{2} \)
7 \( 1 + (0.342 + 0.939i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.642 - 0.766i)T + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-1.50 - 1.26i)T + (0.173 + 0.984i)T^{2} \)
19 \( 1 + (0.642 - 0.766i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.0451 - 0.168i)T + (-0.866 + 0.5i)T^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (1.10 + 1.58i)T + (-0.342 + 0.939i)T^{2} \)
59 \( 1 + (-0.342 + 0.939i)T^{2} \)
61 \( 1 + (-1.98 - 0.173i)T + (0.984 + 0.173i)T^{2} \)
67 \( 1 + (-0.342 - 0.939i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
79 \( 1 + (0.342 + 0.939i)T^{2} \)
83 \( 1 + (-0.984 + 0.173i)T^{2} \)
89 \( 1 + (-0.939 - 1.34i)T + (-0.342 + 0.939i)T^{2} \)
97 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.826752414426866056124977500912, −8.163525693418598977217471321860, −7.78302975465680027591967839312, −6.49273220146175649688765192584, −5.79583919186843143512697013782, −5.17602365526183877746685428889, −4.27072469917316774192959250035, −3.44632894779437287889833616757, −2.15915779752384400567245383998, −1.22965574329752461575909540601, 0.982158062475347556990028231859, 2.63866933735507380083447478737, 3.14912854134855904873253929217, 3.97493441220672547057930830888, 5.32334028108498393853834670827, 5.89091189992771669563637929631, 6.56953246452095243172121574005, 7.50700217127506128082012931581, 7.971229115965478190288429665220, 9.060963961619343071342139498892

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