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Properties

Label 2-960-240.29-c0-0-0
Degree $2$
Conductor $960$
Sign $0.923 - 0.382i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
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  + (−0.707 − 0.707i)3-s + (−0.707 + 0.707i)5-s + 1.00i·9-s + 1.00·15-s + 1.41·17-s + (1 + i)19-s + 1.41i·23-s − 1.00i·25-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)45-s + 1.41·47-s − 49-s + (−1.00 − 1.00i)51-s − 1.41i·57-s + (1 + i)61-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (−0.707 + 0.707i)5-s + 1.00i·9-s + 1.00·15-s + 1.41·17-s + (1 + i)19-s + 1.41i·23-s − 1.00i·25-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)45-s + 1.41·47-s − 49-s + (−1.00 − 1.00i)51-s − 1.41i·57-s + (1 + i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7156592348\)
\(L(\frac12)\) \(\approx\) \(0.7156592348\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + iT^{2} \)
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

−10.36558232986173764903152328891, −9.698155205188167412355580046299, −8.255737972740826183164743737639, −7.58000752147327875751370612629, −7.09789416649322380977365357529, −5.93501165329974293263810987650, −5.34513621981013508106646170895, −3.93439942715813729771767355662, −2.95074215193585422544786187958, −1.37174117917484634184311467975, 0.894775071984483033248944915766, 3.07027421732028272537879621192, 4.07843222763765551901920781336, 4.94039792207487197577294687028, 5.59723686904818904174306294797, 6.75078818536465880981433340165, 7.69185524582873840404655483806, 8.635805100144743904129373147934, 9.412012125410053695396653930178, 10.17185099405675579405742713151

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