Properties

Degree $2$
Conductor $1008$
Sign $0.832 + 0.553i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.35 + 1.93i)5-s + (2.5 − 0.866i)7-s + (−3.35 − 1.93i)11-s − 3.46i·13-s + (2 + 3.46i)19-s + (6.70 − 3.87i)23-s + (5.00 − 8.66i)25-s + 6.70·29-s + (−0.5 + 0.866i)31-s + (−6.70 + 7.74i)35-s + (−2 − 3.46i)37-s − 7.74i·41-s + 6.92i·43-s + (6.70 + 11.6i)47-s + (5.5 − 4.33i)49-s + ⋯
L(s)  = 1  + (−1.50 + 0.866i)5-s + (0.944 − 0.327i)7-s + (−1.01 − 0.583i)11-s − 0.960i·13-s + (0.458 + 0.794i)19-s + (1.39 − 0.807i)23-s + (1.00 − 1.73i)25-s + 1.24·29-s + (−0.0898 + 0.155i)31-s + (−1.13 + 1.30i)35-s + (−0.328 − 0.569i)37-s − 1.20i·41-s + 1.05i·43-s + (0.978 + 1.69i)47-s + (0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.832 + 0.553i$
Motivic weight: \(1\)
Character: $\chi_{1008} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.170293580\)
\(L(\frac12)\) \(\approx\) \(1.170293580\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (3.35 - 1.93i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.35 + 1.93i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.70 + 3.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.74iT - 41T^{2} \)
43 \( 1 - 6.92iT - 43T^{2} \)
47 \( 1 + (-6.70 - 11.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.35 + 5.80i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.35 + 5.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9 + 5.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 3.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.74iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.5 + 6.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + (6.70 - 3.87i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−10.33641221700332814637509654340, −8.745757819282291036192987571804, −7.951159317968439430629295449603, −7.66667485231763518507641664212, −6.71493363026460909838599242864, −5.43928110111558380197361965484, −4.56399411658811557480021688914, −3.48222220852636701031637167404, −2.72337014160572371331039918924, −0.67496184238300226305476616076, 1.10657997650339053144143930908, 2.65570274181617204534961235923, 4.03226309903881700495659169597, 4.81877832650278341383899386361, 5.31310557199032734963821914368, 7.10227121930539293988648035325, 7.47200918790317507316074162516, 8.591784782589994908015626606678, 8.774488850707344848877975080005, 10.04932430371784215239368097236

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