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Properties

Label 2-960-40.37-c2-0-31
Degree $2$
Conductor $960$
Sign $0.363 + 0.931i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s + (3.24 − 3.80i)5-s + (5.40 + 5.40i)7-s + 2.99i·9-s − 2.61i·11-s + (−12.6 − 12.6i)13-s + (−8.63 + 0.695i)15-s + (8.48 + 8.48i)17-s + 24.7·19-s − 13.2i·21-s + (7.02 − 7.02i)23-s + (−4.00 − 24.6i)25-s + (3.67 − 3.67i)27-s + 13.3·29-s + 35.3·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.648 − 0.761i)5-s + (0.772 + 0.772i)7-s + 0.333i·9-s − 0.238i·11-s + (−0.975 − 0.975i)13-s + (−0.575 + 0.0463i)15-s + (0.499 + 0.499i)17-s + 1.30·19-s − 0.630i·21-s + (0.305 − 0.305i)23-s + (−0.160 − 0.987i)25-s + (0.136 − 0.136i)27-s + 0.460·29-s + 1.14·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.363 + 0.931i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.363 + 0.931i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.988780500\)
\(L(\frac12)\) \(\approx\) \(1.988780500\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (-3.24 + 3.80i)T \)
good7 \( 1 + (-5.40 - 5.40i)T + 49iT^{2} \)
11 \( 1 + 2.61iT - 121T^{2} \)
13 \( 1 + (12.6 + 12.6i)T + 169iT^{2} \)
17 \( 1 + (-8.48 - 8.48i)T + 289iT^{2} \)
19 \( 1 - 24.7T + 361T^{2} \)
23 \( 1 + (-7.02 + 7.02i)T - 529iT^{2} \)
29 \( 1 - 13.3T + 841T^{2} \)
31 \( 1 - 35.3T + 961T^{2} \)
37 \( 1 + (12.6 - 12.6i)T - 1.36e3iT^{2} \)
41 \( 1 + 67.6T + 1.68e3T^{2} \)
43 \( 1 + (34.8 + 34.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (-34.4 - 34.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (-49.1 - 49.1i)T + 2.80e3iT^{2} \)
59 \( 1 - 58.9T + 3.48e3T^{2} \)
61 \( 1 + 53.2iT - 3.72e3T^{2} \)
67 \( 1 + (-78.6 + 78.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 31.2T + 5.04e3T^{2} \)
73 \( 1 + (-42.3 + 42.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 8.36iT - 6.24e3T^{2} \)
83 \( 1 + (66.5 + 66.5i)T + 6.88e3iT^{2} \)
89 \( 1 + 108. iT - 7.92e3T^{2} \)
97 \( 1 + (100. + 100. i)T + 9.40e3iT^{2} \)
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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

−9.814348153764934542438605026906, −8.626508651580848815159901460430, −8.172780400021218035776284711117, −7.16852568772402146439185634490, −5.98958039118872888294272800203, −5.29031129109239983959898067142, −4.81293292321546579403382372126, −3.03557701712422215319256998182, −1.89247669446039404378745285797, −0.77040774637395327193228450044, 1.20648215142332117128334469514, 2.53817901477394953712806620983, 3.75179886085610582174701507838, 4.87620761231094029638971801557, 5.44885827059215649700577095311, 6.89169625304245151998837125507, 7.09968104426002380859915595374, 8.279930467058702483289180387070, 9.595754660227365959000330190603, 9.915791376665740465961029815546

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