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Properties

Label 2-2960-740.347-c0-0-0
Degree $2$
Conductor $2960$
Sign $-0.937 + 0.348i$
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 + (−0.866 − 0.5i)9-s + (−1.73 + i)13-s + (0.866 − 1.5i)17-s + (−0.499 − 0.866i)25-s + (−1.36 + 1.36i)29-s + (−0.5 − 0.866i)37-s + (−1.5 + 0.866i)41-s + (0.866 − 0.499i)45-s + (−0.866 − 0.5i)49-s + (−0.366 − 1.36i)53-s + (0.133 − 0.5i)61-s − 1.99i·65-s + (−1 − i)73-s + (0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (−0.866 − 0.5i)9-s + (−1.73 + i)13-s + (0.866 − 1.5i)17-s + (−0.499 − 0.866i)25-s + (−1.36 + 1.36i)29-s + (−0.5 − 0.866i)37-s + (−1.5 + 0.866i)41-s + (0.866 − 0.499i)45-s + (−0.866 − 0.5i)49-s + (−0.366 − 1.36i)53-s + (0.133 − 0.5i)61-s − 1.99i·65-s + (−1 − i)73-s + (0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.937 + 0.348i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ -0.937 + 0.348i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002470173231\)
\(L(\frac12)\) \(\approx\) \(0.002470173231\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + (0.866 + 0.5i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.527048959315649276054476240824, −8.605070413647488236943602487118, −7.64150623038984642023047624095, −7.10147707922960818258433197927, −6.57447549969514994483479263251, −5.39107114501179877737321513556, −4.83272639339818627761784526062, −3.57691299166272726751923612228, −3.01341478936054051873662315044, −2.03618705487495805312813975005, 0.00141906712696950118489913457, 1.67699983410405961918386064045, 2.81118967379193187918222545072, 3.74373285681990508309180291405, 4.72125306797523499504566416506, 5.42541947463384981645629659827, 5.94426764172159660447939560920, 7.29238529736788540997732257865, 7.914617109662855853069759282414, 8.301735981042639559618292713589

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