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Properties

Label 2-1008-7.4-c1-0-8
Degree $2$
Conductor $1008$
Sign $0.968 + 0.250i$
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  + (−0.5 − 0.866i)5-s + (2.5 − 0.866i)7-s + (−2.5 + 4.33i)11-s + 2·13-s + (3 − 5.19i)17-s + (1 + 1.73i)19-s + (3 + 5.19i)23-s + (2 − 3.46i)25-s + 3·29-s + (2.5 − 4.33i)31-s + (−2 − 1.73i)35-s + (1 + 1.73i)37-s − 8·41-s + 4·43-s + (−2 − 3.46i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.944 − 0.327i)7-s + (−0.753 + 1.30i)11-s + 0.554·13-s + (0.727 − 1.26i)17-s + (0.229 + 0.397i)19-s + (0.625 + 1.08i)23-s + (0.400 − 0.692i)25-s + 0.557·29-s + (0.449 − 0.777i)31-s + (−0.338 − 0.292i)35-s + (0.164 + 0.284i)37-s − 1.24·41-s + 0.609·43-s + (−0.291 − 0.505i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.737417722\)
\(L(\frac12)\) \(\approx\) \(1.737417722\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17T + 83T^{2} \)
89 \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3T + 97T^{2} \)
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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

−9.993624471948083077807438654306, −9.124786266611417920648835816286, −8.052040929418026035766767964474, −7.62477930222874823347204976306, −6.70843931860780440560655264254, −5.23117546371561799274185202640, −4.89464998761551061962192720083, −3.75064916702532926742000564267, −2.37988467931317034692843062824, −1.06071389403996790408828165924, 1.15345886965525059972702960911, 2.69585064722555527155242267102, 3.59486018450173281734953727325, 4.86122344928932565519473209028, 5.67203027016121606013357166853, 6.54400323907823576437181993639, 7.66750885795739003211602129956, 8.419703141445039250311568216505, 8.829721428087956030341195948348, 10.30154289759245341865228393438

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