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Properties

Label 2-2646-63.16-c1-0-32
Degree $2$
Conductor $2646$
Sign $0.678 + 0.734i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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)2-s + (−0.499 + 0.866i)4-s − 5-s − 0.999·8-s + (−0.5 − 0.866i)10-s + 2·11-s + (−1 − 1.73i)13-s + (−0.5 − 0.866i)16-s + (3.5 − 6.06i)19-s + (0.499 − 0.866i)20-s + (1 + 1.73i)22-s − 3·23-s − 4·25-s + (0.999 − 1.73i)26-s + (−4 + 6.92i)29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.447·5-s − 0.353·8-s + (−0.158 − 0.273i)10-s + 0.603·11-s + (−0.277 − 0.480i)13-s + (−0.125 − 0.216i)16-s + (0.802 − 1.39i)19-s + (0.111 − 0.193i)20-s + (0.213 + 0.369i)22-s − 0.625·23-s − 0.800·25-s + (0.196 − 0.339i)26-s + (−0.742 + 1.28i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.678 + 0.734i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.281762645\)
\(L(\frac12)\) \(\approx\) \(1.281762645\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (4 - 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.790073489051813547790157071250, −7.77705060785520364731253813385, −7.23146916386085925745256666711, −6.60012444570536897520206732669, −5.50397939311162127705588488747, −5.03961284097042814273162058216, −3.90442341432367531105968979944, −3.36360543624984668151586343742, −2.06199233229338378541278059606, −0.39055283512884791749092814124, 1.25022950695118278142711486912, 2.25103453518593034663524354729, 3.44613036914092216390024424897, 4.04244210939710265462976143361, 4.82504869611682073725806830827, 5.93717072777044510102241792586, 6.39874533411096085081170093180, 7.77794422133771321119184879946, 7.935483617834508452381134875830, 9.277167559303319779577149323383

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