Properties

Label 2-960-192.35-c1-0-127
Degree $2$
Conductor $960$
Sign $-0.0669 - 0.997i$
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  + (0.557 − 1.29i)2-s + (−0.0725 − 1.73i)3-s + (−1.37 − 1.44i)4-s + (−0.831 − 0.555i)5-s + (−2.28 − 0.869i)6-s + (−0.966 − 2.33i)7-s + (−2.65 + 0.986i)8-s + (−2.98 + 0.251i)9-s + (−1.18 + 0.771i)10-s + (0.314 + 1.57i)11-s + (−2.40 + 2.49i)12-s + (−0.595 − 0.890i)13-s + (−3.57 − 0.0434i)14-s + (−0.901 + 1.47i)15-s + (−0.194 + 3.99i)16-s + (3.51 − 3.51i)17-s + ⋯
L(s)  = 1  + (0.393 − 0.919i)2-s + (−0.0418 − 0.999i)3-s + (−0.689 − 0.724i)4-s + (−0.371 − 0.248i)5-s + (−0.934 − 0.355i)6-s + (−0.365 − 0.882i)7-s + (−0.937 + 0.348i)8-s + (−0.996 + 0.0836i)9-s + (−0.374 + 0.243i)10-s + (0.0947 + 0.476i)11-s + (−0.694 + 0.719i)12-s + (−0.165 − 0.247i)13-s + (−0.954 − 0.0116i)14-s + (−0.232 + 0.381i)15-s + (−0.0486 + 0.998i)16-s + (0.852 − 0.852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0669 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0669 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.0669 - 0.997i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.0669 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.553169 + 0.591548i\)
\(L(\frac12)\) \(\approx\) \(0.553169 + 0.591548i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.557 + 1.29i)T \)
3 \( 1 + (0.0725 + 1.73i)T \)
5 \( 1 + (0.831 + 0.555i)T \)
good7 \( 1 + (0.966 + 2.33i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.314 - 1.57i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (0.595 + 0.890i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (-3.51 + 3.51i)T - 17iT^{2} \)
19 \( 1 + (1.27 + 1.91i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (1.44 - 3.48i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.64 - 0.327i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + 4.99T + 31T^{2} \)
37 \( 1 + (0.709 + 0.474i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-0.102 - 0.0423i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-0.442 - 2.22i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (3.58 + 3.58i)T + 47iT^{2} \)
53 \( 1 + (-10.6 + 2.11i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (1.61 - 2.41i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (7.53 + 1.49i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-2.21 + 11.1i)T + (-61.8 - 25.6i)T^{2} \)
71 \( 1 + (5.24 - 2.17i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.27 + 1.77i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (10.8 + 10.8i)T + 79iT^{2} \)
83 \( 1 + (9.64 - 6.44i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (12.1 - 5.01i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 0.469iT - 97T^{2} \)
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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

−9.568944814738313629617892479725, −8.659442238481827492455062875442, −7.59309743657305388271381873633, −6.97742283369873089841835377088, −5.81026792365920215107048656451, −4.91296243418766681777274076216, −3.78352235730910401102643951835, −2.84461214031830055446573015721, −1.52120732084997294800001468777, −0.33768448191008640138407762102, 2.79030318861183820226300288815, 3.69198829259901784585585666312, 4.48798520195742743775694161189, 5.71014093814517424473870600263, 5.99718306217697753949233215931, 7.20593559265409159842144184753, 8.370465203070387295968032845268, 8.710918593792193831292556590566, 9.696831972262177309348233812350, 10.46373111630823678937822861875

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