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Properties

Label 2-1008-252.187-c1-0-37
Degree $2$
Conductor $1008$
Sign $0.237 + 0.971i$
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  + (1.13 − 1.30i)3-s − 0.618i·5-s + (1.03 + 2.43i)7-s + (−0.413 − 2.97i)9-s − 2.97i·11-s + (−1.64 + 0.951i)13-s + (−0.808 − 0.703i)15-s + (2.92 − 1.68i)17-s + (1.13 − 1.96i)19-s + (4.35 + 1.41i)21-s − 6.74i·23-s + 4.61·25-s + (−4.35 − 2.83i)27-s + (−1.74 + 3.02i)29-s + (3.67 − 6.36i)31-s + ⋯
L(s)  = 1  + (0.656 − 0.754i)3-s − 0.276i·5-s + (0.391 + 0.920i)7-s + (−0.137 − 0.990i)9-s − 0.898i·11-s + (−0.457 + 0.263i)13-s + (−0.208 − 0.181i)15-s + (0.708 − 0.409i)17-s + (0.260 − 0.450i)19-s + (0.951 + 0.308i)21-s − 1.40i·23-s + 0.923·25-s + (−0.837 − 0.546i)27-s + (−0.324 + 0.562i)29-s + (0.660 − 1.14i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.237 + 0.971i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.237 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.023419068\)
\(L(\frac12)\) \(\approx\) \(2.023419068\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.13 + 1.30i)T \)
7 \( 1 + (-1.03 - 2.43i)T \)
good5 \( 1 + 0.618iT - 5T^{2} \)
11 \( 1 + 2.97iT - 11T^{2} \)
13 \( 1 + (1.64 - 0.951i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.92 + 1.68i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.13 + 1.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.74iT - 23T^{2} \)
29 \( 1 + (1.74 - 3.02i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.67 + 6.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.25 - 5.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.09 + 5.25i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.48 + 1.43i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.80 - 4.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.50 - 4.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.17 + 3.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.14 - 1.81i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.34 + 2.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.18iT - 71T^{2} \)
73 \( 1 + (12.5 - 7.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.4 + 6.03i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.48 - 9.49i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.03 + 3.48i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.71 - 1.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.435207658236264141209424671159, −8.864141232501041234286600661792, −8.230963986692283669320557840655, −7.40240420027975699883239037182, −6.42031660380850643862974166662, −5.58137465051895647652410947897, −4.52898395297217489008279894917, −3.09800607654435792707232309535, −2.37000173474308964550689502882, −0.926336115524491526202709049133, 1.60749597885102106254979750779, 2.98034726253346908181088895253, 3.89558730402596238315712914722, 4.73828856365889202792008973203, 5.63458650734893734379655600478, 7.14530119211253459128728632672, 7.59006118836072116481467363887, 8.454879174242249226262553477369, 9.532510173012392914929375598556, 10.10343435050274269249653037699

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