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Properties

Label 2-960-960.509-c0-0-1
Degree $2$
Conductor $960$
Sign $-0.881 - 0.471i$
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  + (−0.555 − 0.831i)2-s + (0.195 − 0.980i)3-s + (−0.382 + 0.923i)4-s + (−0.831 − 0.555i)5-s + (−0.923 + 0.382i)6-s + (0.980 − 0.195i)8-s + (−0.923 − 0.382i)9-s + i·10-s + (0.831 + 0.555i)12-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)16-s + (−1.17 − 1.17i)17-s + (0.195 + 0.980i)18-s + (−1.08 − 1.63i)19-s + (0.831 − 0.555i)20-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)2-s + (0.195 − 0.980i)3-s + (−0.382 + 0.923i)4-s + (−0.831 − 0.555i)5-s + (−0.923 + 0.382i)6-s + (0.980 − 0.195i)8-s + (−0.923 − 0.382i)9-s + i·10-s + (0.831 + 0.555i)12-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)16-s + (−1.17 − 1.17i)17-s + (0.195 + 0.980i)18-s + (−1.08 − 1.63i)19-s + (0.831 − 0.555i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.881 - 0.471i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ -0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4308677026\)
\(L(\frac12)\) \(\approx\) \(0.4308677026\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.555 + 0.831i)T \)
3 \( 1 + (-0.195 + 0.980i)T \)
5 \( 1 + (0.831 + 0.555i)T \)
good7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.923 + 0.382i)T^{2} \)
13 \( 1 + (-0.382 + 0.923i)T^{2} \)
17 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
19 \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \)
23 \( 1 + (0.425 - 1.02i)T + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.923 - 0.382i)T^{2} \)
31 \( 1 - 0.765iT - T^{2} \)
37 \( 1 + (0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (0.923 - 0.382i)T^{2} \)
47 \( 1 + (0.275 + 0.275i)T + iT^{2} \)
53 \( 1 + (-1.81 + 0.360i)T + (0.923 - 0.382i)T^{2} \)
59 \( 1 + (-0.382 - 0.923i)T^{2} \)
61 \( 1 + (-0.382 + 1.92i)T + (-0.923 - 0.382i)T^{2} \)
67 \( 1 + (0.923 + 0.382i)T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (1 - i)T - iT^{2} \)
83 \( 1 + (1.02 + 1.53i)T + (-0.382 + 0.923i)T^{2} \)
89 \( 1 + (0.707 - 0.707i)T^{2} \)
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

−9.518536626128761323051034137008, −8.775113251732476809615071764701, −8.368658278364609727640020302453, −7.24718367241857159532103922957, −6.87448716923298657954776950798, −5.16399645929114841594038805442, −4.19143678995969743312446516597, −3.02028299890156370522572771486, −1.99970084154301319681594555516, −0.46847616662544562363956571751, 2.31813725068170469308568262804, 4.06067507860493016992428763165, 4.26816368720449401224860375372, 5.78609970667880574440301822229, 6.40385368956008270993321681978, 7.52899136104683874228029700735, 8.461088471723454456927584806518, 8.664481793618021925428635881449, 9.992805855025713129200206819598, 10.50305472569036736260297163260

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