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Properties

Label 2-960-5.3-c2-0-43
Degree $2$
Conductor $960$
Sign $-0.702 + 0.711i$
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 + (−2.65 − 4.23i)5-s + (0.640 − 0.640i)7-s + 2.99i·9-s + 6.31·11-s + (−6.44 − 6.44i)13-s + (1.93 − 8.44i)15-s + (−7.85 + 7.85i)17-s − 9.85i·19-s + 1.56·21-s + (−10.1 − 10.1i)23-s + (−10.9 + 22.4i)25-s + (−3.67 + 3.67i)27-s − 21.5i·29-s − 0.233·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.530 − 0.847i)5-s + (0.0914 − 0.0914i)7-s + 0.333i·9-s + 0.574·11-s + (−0.495 − 0.495i)13-s + (0.129 − 0.562i)15-s + (−0.462 + 0.462i)17-s − 0.518i·19-s + 0.0746·21-s + (−0.442 − 0.442i)23-s + (−0.436 + 0.899i)25-s + (−0.136 + 0.136i)27-s − 0.742i·29-s − 0.00752·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.702 + 0.711i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ -0.702 + 0.711i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8727040138\)
\(L(\frac12)\) \(\approx\) \(0.8727040138\)
\(L(2)\) 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 \( 1 + (2.65 + 4.23i)T \)
good7 \( 1 + (-0.640 + 0.640i)T - 49iT^{2} \)
11 \( 1 - 6.31T + 121T^{2} \)
13 \( 1 + (6.44 + 6.44i)T + 169iT^{2} \)
17 \( 1 + (7.85 - 7.85i)T - 289iT^{2} \)
19 \( 1 + 9.85iT - 361T^{2} \)
23 \( 1 + (10.1 + 10.1i)T + 529iT^{2} \)
29 \( 1 + 21.5iT - 841T^{2} \)
31 \( 1 + 0.233T + 961T^{2} \)
37 \( 1 + (-11.1 + 11.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 22.0T + 1.68e3T^{2} \)
43 \( 1 + (39.3 + 39.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (19.3 - 19.3i)T - 2.20e3iT^{2} \)
53 \( 1 + (61.2 + 61.2i)T + 2.80e3iT^{2} \)
59 \( 1 + 91.7iT - 3.48e3T^{2} \)
61 \( 1 + 1.60T + 3.72e3T^{2} \)
67 \( 1 + (63.8 - 63.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 120.T + 5.04e3T^{2} \)
73 \( 1 + (-62.8 - 62.8i)T + 5.32e3iT^{2} \)
79 \( 1 + 68.9iT - 6.24e3T^{2} \)
83 \( 1 + (63.8 + 63.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 104. iT - 7.92e3T^{2} \)
97 \( 1 + (-38.5 + 38.5i)T - 9.40e3iT^{2} \)
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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

−9.456982657309816142040269025216, −8.637603195614320483150106633442, −8.065766667582245392577821191739, −7.13447037942510770311852020750, −5.98666323779675185557787551276, −4.85456139984230981802080927017, −4.24700077847743561184947436745, −3.21404157794570278206573128853, −1.81131238376415904695580497393, −0.25346346689475379027468252259, 1.61858779223939276106665987173, 2.79564720525715068328374027717, 3.71722151903109365374896262314, 4.74058599239633751926806865251, 6.12826941315012304820956068624, 6.86777668388770620865033576160, 7.54920943135955738413244761857, 8.379663316112248781192555600869, 9.292707492240419869812049160368, 10.08094680480193586813692184836

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