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Properties

Label 2-960-120.29-c2-0-36
Degree $2$
Conductor $960$
Sign $0.985 - 0.171i$
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  + (2.54 − 1.59i)3-s + (−4.96 − 0.625i)5-s + 5.93i·7-s + (3.92 − 8.10i)9-s + 5.99·11-s − 15.6·13-s + (−13.6 + 6.31i)15-s + 14.2·17-s + 29.6i·19-s + (9.46 + 15.0i)21-s + 21.5·23-s + (24.2 + 6.20i)25-s + (−2.93 − 26.8i)27-s + 18.1·29-s + 8.55·31-s + ⋯
L(s)  = 1  + (0.847 − 0.531i)3-s + (−0.992 − 0.125i)5-s + 0.848i·7-s + (0.435 − 0.900i)9-s + 0.545·11-s − 1.20·13-s + (−0.907 + 0.420i)15-s + 0.836·17-s + 1.55i·19-s + (0.450 + 0.718i)21-s + 0.935·23-s + (0.968 + 0.248i)25-s + (−0.108 − 0.994i)27-s + 0.625·29-s + 0.276·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.985 - 0.171i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.985 - 0.171i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.131685140\)
\(L(\frac12)\) \(\approx\) \(2.131685140\)
\(L(2)\) 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 \( 1 + (-2.54 + 1.59i)T \)
5 \( 1 + (4.96 + 0.625i)T \)
good7 \( 1 - 5.93iT - 49T^{2} \)
11 \( 1 - 5.99T + 121T^{2} \)
13 \( 1 + 15.6T + 169T^{2} \)
17 \( 1 - 14.2T + 289T^{2} \)
19 \( 1 - 29.6iT - 361T^{2} \)
23 \( 1 - 21.5T + 529T^{2} \)
29 \( 1 - 18.1T + 841T^{2} \)
31 \( 1 - 8.55T + 961T^{2} \)
37 \( 1 - 13.4T + 1.36e3T^{2} \)
41 \( 1 + 18.8iT - 1.68e3T^{2} \)
43 \( 1 - 27.1T + 1.84e3T^{2} \)
47 \( 1 - 37.4T + 2.20e3T^{2} \)
53 \( 1 - 93.5iT - 2.80e3T^{2} \)
59 \( 1 + 83.7T + 3.48e3T^{2} \)
61 \( 1 + 39.3iT - 3.72e3T^{2} \)
67 \( 1 - 103.T + 4.48e3T^{2} \)
71 \( 1 + 14.5iT - 5.04e3T^{2} \)
73 \( 1 - 95.8iT - 5.32e3T^{2} \)
79 \( 1 - 133.T + 6.24e3T^{2} \)
83 \( 1 + 116. iT - 6.88e3T^{2} \)
89 \( 1 - 146. iT - 7.92e3T^{2} \)
97 \( 1 - 19.9iT - 9.40e3T^{2} \)
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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

−9.567539978197103954550558636692, −8.971460545361713813915560694490, −8.045968067090824785741229519033, −7.58624801937744895799586134904, −6.65365360708340467541092227899, −5.54164648437120940890208846070, −4.34704543956794528769526233076, −3.37931308847122401994962412081, −2.45683851600493324400674260754, −1.06161085162615673328934122152, 0.77218672301478848782184096327, 2.61102440015062874180171115448, 3.46337636852286945904423755628, 4.42242210798422530535790330362, 4.99685937578550930111692313163, 6.82169748954287405697265300915, 7.35863721989760389046058785519, 8.080255934771413197791931942135, 9.028600910685971717295948959979, 9.726747015386904420167000819641

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