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Properties

Label 2-2960-740.203-c0-0-0
Degree $2$
Conductor $2960$
Sign $0.402 + 0.915i$
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  + (−0.866 + 0.5i)5-s + (−0.984 − 0.173i)9-s + (−0.173 − 0.984i)13-s + (0.673 + 0.118i)17-s + (0.499 − 0.866i)25-s + (−0.424 − 1.58i)29-s + (0.984 − 0.173i)37-s + (0.673 − 0.118i)41-s + (0.939 − 0.342i)45-s + (0.642 − 0.766i)49-s + (−1.75 − 0.816i)53-s + (0.657 − 0.939i)61-s + (0.642 + 0.766i)65-s + (0.366 − 0.366i)73-s + (0.939 + 0.342i)81-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)5-s + (−0.984 − 0.173i)9-s + (−0.173 − 0.984i)13-s + (0.673 + 0.118i)17-s + (0.499 − 0.866i)25-s + (−0.424 − 1.58i)29-s + (0.984 − 0.173i)37-s + (0.673 − 0.118i)41-s + (0.939 − 0.342i)45-s + (0.642 − 0.766i)49-s + (−1.75 − 0.816i)53-s + (0.657 − 0.939i)61-s + (0.642 + 0.766i)65-s + (0.366 − 0.366i)73-s + (0.939 + 0.342i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.402 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7707880606\)
\(L(\frac12)\) \(\approx\) \(0.7707880606\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (-0.984 + 0.173i)T \)
good3 \( 1 + (0.984 + 0.173i)T^{2} \)
7 \( 1 + (-0.642 + 0.766i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
19 \( 1 + (-0.984 - 0.173i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.424 + 1.58i)T + (-0.866 + 0.5i)T^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (-0.673 + 0.118i)T + (0.939 - 0.342i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (1.75 + 0.816i)T + (0.642 + 0.766i)T^{2} \)
59 \( 1 + (-0.642 - 0.766i)T^{2} \)
61 \( 1 + (-0.657 + 0.939i)T + (-0.342 - 0.939i)T^{2} \)
67 \( 1 + (0.642 - 0.766i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
79 \( 1 + (0.642 - 0.766i)T^{2} \)
83 \( 1 + (-0.342 + 0.939i)T^{2} \)
89 \( 1 + (-0.766 + 1.64i)T + (-0.642 - 0.766i)T^{2} \)
97 \( 1 + (0.300 - 0.173i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.604911440114012857391723993026, −7.906649037658951731900987770097, −7.57272155959841837638966622559, −6.42762203956690888761875959634, −5.82979837982597406458289136114, −4.93834174646304138737201933549, −3.87785946671106035188434009773, −3.19126674052606758907263105876, −2.37152339618070847336196991265, −0.52623751594222488802288036216, 1.24679056425560274760178621453, 2.62969227238084576813300886875, 3.52141546440973922557933605338, 4.40551735910827511086945418157, 5.15475002081975378874749642773, 5.94975073350595764114892035243, 6.92480500430512532332618167576, 7.67084538768811869002760881019, 8.279433345698490357068882949102, 9.083060772106363498495919912866

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