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Properties

Label 2-2960-740.739-c0-0-5
Degree $2$
Conductor $2960$
Sign $1$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.84·3-s − 5-s − 0.765·7-s + 2.41·9-s + 1.41·13-s − 1.84·15-s − 1.41·17-s + 1.84·19-s − 1.41·21-s + 25-s + 2.61·27-s − 0.765·31-s + 0.765·35-s + 37-s + 2.61·39-s − 2.41·45-s + 0.765·47-s − 0.414·49-s − 2.61·51-s + 3.41·57-s − 0.765·59-s − 1.84·63-s − 1.41·65-s − 1.84·67-s + 1.84·75-s + 0.765·79-s + 2.41·81-s + ⋯
L(s)  = 1  + 1.84·3-s − 5-s − 0.765·7-s + 2.41·9-s + 1.41·13-s − 1.84·15-s − 1.41·17-s + 1.84·19-s − 1.41·21-s + 25-s + 2.61·27-s − 0.765·31-s + 0.765·35-s + 37-s + 2.61·39-s − 2.41·45-s + 0.765·47-s − 0.414·49-s − 2.61·51-s + 3.41·57-s − 0.765·59-s − 1.84·63-s − 1.41·65-s − 1.84·67-s + 1.84·75-s + 0.765·79-s + 2.41·81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2959, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.002875741\)
\(L(\frac12)\) \(\approx\) \(2.002875741\)
\(L(1)\) 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 \( 1 + T \)
37 \( 1 - T \)
good3 \( 1 - 1.84T + T^{2} \)
7 \( 1 + 0.765T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.41T + T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 - 1.84T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 0.765T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 0.765T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 0.765T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.84T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 0.765T + T^{2} \)
83 \( 1 + 1.84T + T^{2} \)
89 \( 1 - T^{2} \)
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

−9.008283101554095047823040508908, −8.234238564502130806189935013630, −7.55014283464593365918284353840, −7.02353380719364773441731513123, −6.08901120087902065114952433881, −4.65956509553633570492550124523, −3.86021285792515153135185902792, −3.34621182423389083777388520277, −2.65681745254138831216974915311, −1.33010157346009607046084894090, 1.33010157346009607046084894090, 2.65681745254138831216974915311, 3.34621182423389083777388520277, 3.86021285792515153135185902792, 4.65956509553633570492550124523, 6.08901120087902065114952433881, 7.02353380719364773441731513123, 7.55014283464593365918284353840, 8.234238564502130806189935013630, 9.008283101554095047823040508908

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