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Properties

Label 2-960-15.8-c1-0-13
Degree $2$
Conductor $960$
Sign $0.501 + 0.865i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 − 0.292i)3-s + (−2.12 − 0.707i)5-s + (−3 + 3i)7-s + (2.82 + i)9-s + 1.41i·11-s + (3.41 + 1.82i)15-s + (−4.24 − 4.24i)17-s + 4i·19-s + (5.99 − 4.24i)21-s + (2.82 − 2.82i)23-s + (3.99 + 3i)25-s + (−4.53 − 2.53i)27-s + 1.41·29-s + 2·31-s + (0.414 − 2.41i)33-s + ⋯
L(s)  = 1  + (−0.985 − 0.169i)3-s + (−0.948 − 0.316i)5-s + (−1.13 + 1.13i)7-s + (0.942 + 0.333i)9-s + 0.426i·11-s + (0.881 + 0.472i)15-s + (−1.02 − 1.02i)17-s + 0.917i·19-s + (1.30 − 0.925i)21-s + (0.589 − 0.589i)23-s + (0.799 + 0.600i)25-s + (−0.872 − 0.487i)27-s + 0.262·29-s + 0.359·31-s + (0.0721 − 0.420i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.501 + 0.865i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.501 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438309 - 0.252487i\)
\(L(\frac12)\) \(\approx\) \(0.438309 - 0.252487i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.292i)T \)
5 \( 1 + (2.12 + 0.707i)T \)
good7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (4.24 + 4.24i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-2 + 2i)T - 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (2 + 2i)T + 43iT^{2} \)
47 \( 1 + (5.65 + 5.65i)T + 47iT^{2} \)
53 \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (-4 + 4i)T - 67iT^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + (-3 - 3i)T + 73iT^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 + (-2.82 + 2.82i)T - 83iT^{2} \)
89 \( 1 - 2.82T + 89T^{2} \)
97 \( 1 + (13 - 13i)T - 97iT^{2} \)
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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

−9.892274936371371325401027183752, −9.102210046522541173937664636830, −8.252262747377406486738794086000, −7.06938896224814798396282520958, −6.58186051407385800592714837063, −5.50889841772563668761121048500, −4.74750248704763872280416186836, −3.64227528885171268176814853252, −2.31265892158115896234914469375, −0.38642477405807439032525964120, 0.846896504856356869822266866550, 3.09350537718870851245971435149, 4.01017026160896174296058469157, 4.69064310116119664940246606914, 6.12261449022292716062850930692, 6.74420875306646919625230053952, 7.33716782654360517226552610648, 8.458166373582022874114520876746, 9.572817598701679752627300948692, 10.33658598894866096321725878646

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