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Properties

Label 2-1008-252.191-c1-0-36
Degree $2$
Conductor $1008$
Sign $0.560 + 0.828i$
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  + (1.64 + 0.533i)3-s − 3.05i·5-s + (0.624 − 2.57i)7-s + (2.43 + 1.75i)9-s + 1.38·11-s + (1.50 + 2.61i)13-s + (1.62 − 5.02i)15-s + (−1.45 + 0.841i)17-s + (−6.15 − 3.55i)19-s + (2.40 − 3.90i)21-s + 6.91·23-s − 4.31·25-s + (3.06 + 4.19i)27-s + (1.82 + 1.05i)29-s + (−5.52 − 3.18i)31-s + ⋯
L(s)  = 1  + (0.951 + 0.308i)3-s − 1.36i·5-s + (0.236 − 0.971i)7-s + (0.810 + 0.586i)9-s + 0.417·11-s + (0.418 + 0.724i)13-s + (0.420 − 1.29i)15-s + (−0.353 + 0.204i)17-s + (−1.41 − 0.815i)19-s + (0.524 − 0.851i)21-s + 1.44·23-s − 0.862·25-s + (0.590 + 0.807i)27-s + (0.338 + 0.195i)29-s + (−0.992 − 0.572i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.560 + 0.828i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.560 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.326976700\)
\(L(\frac12)\) \(\approx\) \(2.326976700\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 0.533i)T \)
7 \( 1 + (-0.624 + 2.57i)T \)
good5 \( 1 + 3.05iT - 5T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
13 \( 1 + (-1.50 - 2.61i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.45 - 0.841i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.15 + 3.55i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.91T + 23T^{2} \)
29 \( 1 + (-1.82 - 1.05i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.52 + 3.18i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.02 + 8.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.51 + 2.60i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.80 + 2.77i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.23 + 5.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.693 - 0.400i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.22 - 9.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.73 - 9.93i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.98 - 1.14i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.70T + 71T^{2} \)
73 \( 1 + (-2.75 - 4.77i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.90 - 2.25i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.80 - 8.32i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-16.2 - 9.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.47 - 6.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.511899880307523133324248054744, −8.902901410728419972293950326389, −8.512089141995410893270802910733, −7.42634230890839351901484615486, −6.66353753252811999541747550017, −5.17290250828789333304854665593, −4.31080888432226063666968482099, −3.88257907184901906447436675011, −2.21730808335786322204214181538, −1.04242800815382209011044102665, 1.75917677255339821117515252872, 2.82932837555467933249976725308, 3.40411675759167720456447212143, 4.74960365356233816946824358160, 6.23664628421556994084096013967, 6.60435529664225803706729526293, 7.71274442013419257735248895191, 8.392045638034944872055647875563, 9.159250380055572853760494950207, 10.03882564004521094013770264306

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