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Properties

Label 2-2960-2960.2859-c0-0-1
Degree $2$
Conductor $2960$
Sign $0.969 + 0.243i$
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  + (0.965 − 0.258i)2-s + (0.965 + 0.258i)3-s + (0.866 − 0.499i)4-s + (0.965 + 0.258i)5-s + 6-s + (−1.22 + 0.707i)7-s + (0.707 − 0.707i)8-s + 10-s + (1 − i)11-s + (0.965 − 0.258i)12-s + (0.258 − 0.965i)13-s + (−0.999 + i)14-s + (0.866 + 0.499i)15-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.965 + 0.258i)3-s + (0.866 − 0.499i)4-s + (0.965 + 0.258i)5-s + 6-s + (−1.22 + 0.707i)7-s + (0.707 − 0.707i)8-s + 10-s + (1 − i)11-s + (0.965 − 0.258i)12-s + (0.258 − 0.965i)13-s + (−0.999 + i)14-s + (0.866 + 0.499i)15-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.969 + 0.243i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.969 + 0.243i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.219089349\)
\(L(\frac12)\) \(\approx\) \(3.219089349\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
good3 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1 + i)T - iT^{2} \)
13 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - iT - T^{2} \)
41 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−9.040547132685159259733299452788, −8.387380938418681074760457323749, −7.11229791026891157280496313906, −6.29016949455628464090618409392, −5.93692471231811940268287490446, −5.16808085924188456514195984792, −3.64203260469956747632775179996, −3.37372089916891360089485914056, −2.65800203390311993806934389188, −1.64905434566287743034534347852, 1.87080098186009620701592786835, 2.35229586414907195073418928406, 3.42590095139242383781322223446, 4.24989490136653650199081294486, 4.85847493538260941558489579858, 6.26847176308830703708525223118, 6.68790440974132857642043205056, 6.96494305164395984665273427364, 8.250111044914577416466102962706, 9.000363429623346538915892409915

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