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Properties

Label 2-1008-12.11-c3-0-11
Degree $2$
Conductor $1008$
Sign $0.0917 - 0.995i$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.3i·5-s − 7i·7-s − 38.1·11-s + 73.5·13-s − 33.1i·17-s + 65.0i·19-s + 3.43·23-s + 17.6·25-s − 133. i·29-s + 46.7i·31-s + 72.5·35-s + 69.9·37-s − 18.1i·41-s + 311. i·43-s + 337.·47-s + ⋯
L(s)  = 1  + 0.926i·5-s − 0.377i·7-s − 1.04·11-s + 1.56·13-s − 0.472i·17-s + 0.784i·19-s + 0.0311·23-s + 0.141·25-s − 0.857i·29-s + 0.270i·31-s + 0.350·35-s + 0.310·37-s − 0.0691i·41-s + 1.10i·43-s + 1.04·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.0917 - 0.995i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ 0.0917 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.762201234\)
\(L(\frac12)\) \(\approx\) \(1.762201234\)
\(L(\frac{5}{2})\) 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 \( 1 + 7iT \)
good5 \( 1 - 10.3iT - 125T^{2} \)
11 \( 1 + 38.1T + 1.33e3T^{2} \)
13 \( 1 - 73.5T + 2.19e3T^{2} \)
17 \( 1 + 33.1iT - 4.91e3T^{2} \)
19 \( 1 - 65.0iT - 6.85e3T^{2} \)
23 \( 1 - 3.43T + 1.21e4T^{2} \)
29 \( 1 + 133. iT - 2.43e4T^{2} \)
31 \( 1 - 46.7iT - 2.97e4T^{2} \)
37 \( 1 - 69.9T + 5.06e4T^{2} \)
41 \( 1 + 18.1iT - 6.89e4T^{2} \)
43 \( 1 - 311. iT - 7.95e4T^{2} \)
47 \( 1 - 337.T + 1.03e5T^{2} \)
53 \( 1 - 507. iT - 1.48e5T^{2} \)
59 \( 1 + 426.T + 2.05e5T^{2} \)
61 \( 1 - 787.T + 2.26e5T^{2} \)
67 \( 1 - 596. iT - 3.00e5T^{2} \)
71 \( 1 + 1.12e3T + 3.57e5T^{2} \)
73 \( 1 + 315.T + 3.89e5T^{2} \)
79 \( 1 + 514. iT - 4.93e5T^{2} \)
83 \( 1 - 184.T + 5.71e5T^{2} \)
89 \( 1 - 884. iT - 7.04e5T^{2} \)
97 \( 1 - 945.T + 9.12e5T^{2} \)
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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

−10.00287422703836558951687873933, −8.901674023421559656108705715736, −7.996517754837048833409937395707, −7.32764104797947915372754000732, −6.34004245662857373354396365542, −5.67208584139043369212220298467, −4.39030695573846122361369395949, −3.40292531037214685047166413455, −2.52841164448903099675718324833, −1.07025848725723978622482005221, 0.50532027756097197148422344310, 1.68231584516339634521771436357, 2.98008922830303400203311646591, 4.11632123394703007935612086865, 5.13465264746078574987741559891, 5.77518880834498725264018741198, 6.81567584116680157294295215228, 7.943045429317226540283288635656, 8.665975115276682928796197727035, 9.096766200405138710994596122553

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