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Properties

Label 2-1008-21.17-c3-0-28
Degree $2$
Conductor $1008$
Sign $0.361 + 0.932i$
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  + (−5.74 − 9.94i)5-s + (−17.2 − 6.73i)7-s + (49.6 + 28.6i)11-s + 9.20i·13-s + (−14.5 + 25.1i)17-s + (32.6 − 18.8i)19-s + (−7.27 + 4.19i)23-s + (−3.41 + 5.91i)25-s + 62.3i·29-s + (48.8 + 28.2i)31-s + (32.1 + 210. i)35-s + (146. + 252. i)37-s − 54.2·41-s + 438.·43-s + (−128. − 222. i)47-s + ⋯
L(s)  = 1  + (−0.513 − 0.889i)5-s + (−0.931 − 0.363i)7-s + (1.36 + 0.785i)11-s + 0.196i·13-s + (−0.207 + 0.359i)17-s + (0.393 − 0.227i)19-s + (−0.0659 + 0.0380i)23-s + (−0.0273 + 0.0473i)25-s + 0.399i·29-s + (0.283 + 0.163i)31-s + (0.155 + 1.01i)35-s + (0.649 + 1.12i)37-s − 0.206·41-s + 1.55·43-s + (−0.398 − 0.690i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.361 + 0.932i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ 0.361 + 0.932i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.541695231\)
\(L(\frac12)\) \(\approx\) \(1.541695231\)
\(L(\frac{5}{2})\) 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 \( 1 + (17.2 + 6.73i)T \)
good5 \( 1 + (5.74 + 9.94i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-49.6 - 28.6i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 9.20iT - 2.19e3T^{2} \)
17 \( 1 + (14.5 - 25.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-32.6 + 18.8i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (7.27 - 4.19i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 62.3iT - 2.43e4T^{2} \)
31 \( 1 + (-48.8 - 28.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-146. - 252. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 54.2T + 6.89e4T^{2} \)
43 \( 1 - 438.T + 7.95e4T^{2} \)
47 \( 1 + (128. + 222. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (515. + 297. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-238. + 412. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (548. - 316. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-308. + 534. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 396. iT - 3.57e5T^{2} \)
73 \( 1 + (-39.8 - 23.0i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (344. + 597. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.28e3T + 5.71e5T^{2} \)
89 \( 1 + (595. + 1.03e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 946. iT - 9.12e5T^{2} \)
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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

−9.383244811549438197345488261422, −8.750968774639693878738604391992, −7.74977560107979719918490587273, −6.82358245095959415292268876678, −6.19504544741967505679440650968, −4.80732823766058598147321346143, −4.17266580022615690547129079211, −3.22997523736845187349276665646, −1.63276299228576928191303525789, −0.52589696954511338039139647701, 0.873292602661905979376130525881, 2.59209183773638841597265117409, 3.41013678994047062238559255616, 4.17580050256549220382476017864, 5.72019733074525057614834063058, 6.37477217581592767418827314221, 7.11444492560920424829762283696, 8.006752033072516962530347839605, 9.139676036060404477074466832550, 9.541052351170910334379499589367

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