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Properties

Label 2-1008-63.25-c1-0-9
Degree $2$
Conductor $1008$
Sign $-0.638 - 0.769i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
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  + (0.796 + 1.53i)3-s + (0.230 − 0.398i)5-s + (−0.0665 + 2.64i)7-s + (−1.73 + 2.45i)9-s + (−1.82 − 3.15i)11-s + (0.730 + 1.26i)13-s + (0.796 + 0.0363i)15-s + (−1.86 + 3.23i)17-s + (2.02 + 3.51i)19-s + (−4.12 + 2.00i)21-s + (0.566 − 0.981i)23-s + (2.39 + 4.14i)25-s + (−5.14 − 0.708i)27-s + (−4.48 + 7.77i)29-s + 0.514·31-s + ⋯
L(s)  = 1  + (0.460 + 0.887i)3-s + (0.102 − 0.178i)5-s + (−0.0251 + 0.999i)7-s + (−0.576 + 0.816i)9-s + (−0.549 − 0.952i)11-s + (0.202 + 0.350i)13-s + (0.205 + 0.00938i)15-s + (−0.452 + 0.784i)17-s + (0.465 + 0.805i)19-s + (−0.899 + 0.437i)21-s + (0.118 − 0.204i)23-s + (0.478 + 0.829i)25-s + (−0.990 − 0.136i)27-s + (−0.833 + 1.44i)29-s + 0.0924·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.638 - 0.769i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.638 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.485222920\)
\(L(\frac12)\) \(\approx\) \(1.485222920\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.796 - 1.53i)T \)
7 \( 1 + (0.0665 - 2.64i)T \)
good5 \( 1 + (-0.230 + 0.398i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.82 + 3.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.730 - 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.86 - 3.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.02 - 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.566 + 0.981i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.48 - 7.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.514T + 31T^{2} \)
37 \( 1 + (4.55 + 7.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.472 + 0.819i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.66 - 8.07i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.32T + 47T^{2} \)
53 \( 1 + (-6.21 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 - 2.32T + 67T^{2} \)
71 \( 1 + 1.67T + 71T^{2} \)
73 \( 1 + (6.62 - 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.00T + 79T^{2} \)
83 \( 1 + (3.32 - 5.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.36 + 2.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.59 - 9.68i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.18723838706135580222466634846, −9.329856283384574363270827683250, −8.607414066343825346063273821200, −8.228728891077519426617115939649, −6.87070760134070692327946040449, −5.54162164774283750785680254642, −5.33002620918096334115765300154, −3.88964880745718821979782110897, −3.11528963097026558462445110712, −1.91767305293625226963001913516, 0.62034208852490406629821499416, 2.10317197255407508664578916017, 3.07909097557052223190310110335, 4.28111835877467987750646258256, 5.34905806017071339235062978138, 6.64517518483221519922721365052, 7.13441404492412241055501512863, 7.82842583866908195924399655605, 8.717574429256265248203126541449, 9.731110243448423871396854954894

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