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Properties

Label 2-960-960.419-c1-0-173
Degree $2$
Conductor $960$
Sign $0.263 + 0.964i$
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  + (1.41 + 0.0989i)2-s + (0.337 − 1.69i)3-s + (1.98 + 0.279i)4-s + (1.24 − 1.85i)5-s + (0.644 − 2.36i)6-s + (2.76 + 0.589i)8-s + (−2.77 − 1.14i)9-s + (1.93 − 2.5i)10-s + (1.14 − 3.26i)12-s + (−2.73 − 2.73i)15-s + (3.84 + 1.10i)16-s + (2.31 + 2.31i)17-s + (−3.79 − 1.89i)18-s + (−1.34 − 2.00i)19-s + (2.97 − 3.33i)20-s + ⋯
L(s)  = 1  + (0.997 + 0.0699i)2-s + (0.195 − 0.980i)3-s + (0.990 + 0.139i)4-s + (0.555 − 0.831i)5-s + (0.263 − 0.964i)6-s + (0.978 + 0.208i)8-s + (−0.923 − 0.382i)9-s + (0.612 − 0.790i)10-s + (0.330 − 0.943i)12-s + (−0.707 − 0.707i)15-s + (0.961 + 0.276i)16-s + (0.561 + 0.561i)17-s + (−0.894 − 0.446i)18-s + (−0.307 − 0.460i)19-s + (0.666 − 0.745i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.75309 - 2.10080i\)
\(L(\frac12)\) \(\approx\) \(2.75309 - 2.10080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0989i)T \)
3 \( 1 + (-0.337 + 1.69i)T \)
5 \( 1 + (-1.24 + 1.85i)T \)
good7 \( 1 + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (-2.31 - 2.31i)T + 17iT^{2} \)
19 \( 1 + (1.34 + 2.00i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (-1.92 - 0.799i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (26.7 + 11.0i)T^{2} \)
31 \( 1 + 4.42T + 31T^{2} \)
37 \( 1 + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (5.97 - 5.97i)T - 47iT^{2} \)
53 \( 1 + (-0.228 - 1.14i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (-15.2 - 3.04i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (11.8 + 11.8i)T + 79iT^{2} \)
83 \( 1 + (-5.58 - 8.35i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.854891395262171174365548405548, −8.815679041238631389437582872462, −8.033764413342267114693996630176, −7.17721211868070533647395455404, −6.25653355009562796057921620342, −5.60867043560375489909024595600, −4.68986355259995811128975549417, −3.43963471677366223250174086654, −2.28156112063759601433669430723, −1.27494572958891785854828597971, 2.07516665204833600499586374452, 3.10614066749905843407319478279, 3.80347669947898821717991544728, 4.99325037841928928968256077679, 5.64643095676681212347430800677, 6.57766763782133050287449958235, 7.47246856883702120774546212817, 8.579450116374512151795756873761, 9.828174322612933285056905724830, 10.14210286153092537564455440271

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