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Properties

Label 2-1008-252.31-c1-0-41
Degree $2$
Conductor $1008$
Sign $-0.539 + 0.842i$
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  + (0.840 − 1.51i)3-s − 2.34i·5-s + (2.61 − 0.373i)7-s + (−1.58 − 2.54i)9-s − 0.442i·11-s + (−1.96 − 1.13i)13-s + (−3.55 − 1.97i)15-s + (4.01 + 2.32i)17-s + (−3.30 − 5.73i)19-s + (1.63 − 4.28i)21-s + 1.13i·23-s − 0.514·25-s + (−5.18 + 0.267i)27-s + (4.37 + 7.58i)29-s + (−2.32 − 4.01i)31-s + ⋯
L(s)  = 1  + (0.485 − 0.874i)3-s − 1.05i·5-s + (0.989 − 0.141i)7-s + (−0.529 − 0.848i)9-s − 0.133i·11-s + (−0.543 − 0.314i)13-s + (−0.918 − 0.509i)15-s + (0.974 + 0.562i)17-s + (−0.759 − 1.31i)19-s + (0.356 − 0.934i)21-s + 0.236i·23-s − 0.102·25-s + (−0.998 + 0.0515i)27-s + (0.812 + 1.40i)29-s + (−0.416 − 0.721i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.539 + 0.842i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.539 + 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.920895874\)
\(L(\frac12)\) \(\approx\) \(1.920895874\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.840 + 1.51i)T \)
7 \( 1 + (-2.61 + 0.373i)T \)
good5 \( 1 + 2.34iT - 5T^{2} \)
11 \( 1 + 0.442iT - 11T^{2} \)
13 \( 1 + (1.96 + 1.13i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.01 - 2.32i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.30 + 5.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.13iT - 23T^{2} \)
29 \( 1 + (-4.37 - 7.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.32 + 4.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.41 + 2.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.55 - 2.62i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.81 - 2.20i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.67 - 4.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.54 - 2.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.689 + 1.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.88 + 5.13i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.10 + 1.21i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.20iT - 71T^{2} \)
73 \( 1 + (-3.17 - 1.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.26 - 4.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.87 + 13.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.32 + 3.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.9 + 8.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.340011702950142830336197411289, −8.747040027341838490365993310078, −8.015881922716764302385312502744, −7.42980299495439877718824724409, −6.33159900830591290745743962870, −5.24957884952684272222528940887, −4.52737822673152156949310466722, −3.15067882167686014764604458363, −1.87109750159843865833544747191, −0.856887634041106552212204012146, 2.00699989782905030023107826272, 2.96967904113476592325491842528, 4.01151164497447863732028963719, 4.91000890442890078047368194383, 5.84588792245026642328206651753, 7.01933283097486883172921652261, 7.905879943679697124542325822206, 8.493133870226677635929990563177, 9.615620031152461354943804477765, 10.26482060265230880774690007999

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