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Properties

Label 2-2646-9.7-c1-0-10
Degree $2$
Conductor $2646$
Sign $-0.0871 - 0.996i$
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 + (0.258 + 0.448i)5-s + 0.999·8-s − 0.517·10-s + (−0.732 + 1.26i)11-s + (1.22 + 2.12i)13-s + (−0.5 + 0.866i)16-s + 3.48·17-s + 0.517·19-s + (0.258 − 0.448i)20-s + (−0.732 − 1.26i)22-s + (3.96 + 6.86i)23-s + (2.36 − 4.09i)25-s − 2.44·26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.115 + 0.200i)5-s + 0.353·8-s − 0.163·10-s + (−0.220 + 0.382i)11-s + (0.339 + 0.588i)13-s + (−0.125 + 0.216i)16-s + 0.845·17-s + 0.118·19-s + (0.0578 − 0.100i)20-s + (−0.156 − 0.270i)22-s + (0.826 + 1.43i)23-s + (0.473 − 0.819i)25-s − 0.480·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.0871 - 0.996i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.0871 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.429792326\)
\(L(\frac12)\) \(\approx\) \(1.429792326\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.448i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.732 - 1.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.48T + 17T^{2} \)
19 \( 1 - 0.517T + 19T^{2} \)
23 \( 1 + (-3.96 - 6.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.36 + 2.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.67 + 6.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.09 + 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.31 - 4.00i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.73T + 53T^{2} \)
59 \( 1 + (-7.39 - 12.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.19 - 3.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.90 - 3.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.803T + 71T^{2} \)
73 \( 1 - 4.62T + 73T^{2} \)
79 \( 1 + (7.06 - 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.94 - 8.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + (-0.517 + 0.896i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.063051486017162941692768949138, −8.247084356378315908000078934792, −7.43085083265047523026293199362, −6.96828448545587040373086792994, −5.97114680534973446552773085517, −5.39565484055828799043449480889, −4.41639973142503509726168742615, −3.48217886823819974524648944314, −2.27223498255518410176836390058, −1.08115516723396993399930218732, 0.63924779108701704838973630761, 1.69203459382976939253859355175, 3.00846790339865001767687219059, 3.47107837683834907808804272296, 4.79191853174913357865734027324, 5.35577343986985624906599490495, 6.41285684052566817730929944226, 7.27791376296272286945914777591, 8.101282947201177893458974135789, 8.758815387151240867752788965655

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