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Properties

Label 2-2664-37.26-c1-0-25
Degree $2$
Conductor $2664$
Sign $0.993 - 0.116i$
Analytic cond. $21.2721$
Root an. cond. $4.61217$
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  + (0.717 − 1.24i)5-s + (−0.0542 + 0.0938i)7-s + 4.03·11-s + (−1.19 + 2.07i)13-s + (−0.5 − 0.866i)17-s + (−1.82 + 3.15i)19-s + 3.53·23-s + (1.46 + 2.54i)25-s − 1.30·29-s + 4.03·31-s + (0.0778 + 0.134i)35-s + (5.87 − 1.58i)37-s + (−1.53 + 2.65i)41-s + 2.43·43-s − 5.40·47-s + ⋯
L(s)  = 1  + (0.321 − 0.556i)5-s + (−0.0204 + 0.0354i)7-s + 1.21·11-s + (−0.332 + 0.575i)13-s + (−0.121 − 0.210i)17-s + (−0.417 + 0.723i)19-s + 0.736·23-s + (0.293 + 0.509i)25-s − 0.242·29-s + 0.725·31-s + (0.0131 + 0.0227i)35-s + (0.965 − 0.260i)37-s + (−0.239 + 0.414i)41-s + 0.371·43-s − 0.788·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $0.993 - 0.116i$
Analytic conductor: \(21.2721\)
Root analytic conductor: \(4.61217\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :1/2),\ 0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.066735234\)
\(L(\frac12)\) \(\approx\) \(2.066735234\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37 \( 1 + (-5.87 + 1.58i)T \)
good5 \( 1 + (-0.717 + 1.24i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.0542 - 0.0938i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.03T + 11T^{2} \)
13 \( 1 + (1.19 - 2.07i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.82 - 3.15i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.53T + 23T^{2} \)
29 \( 1 + 1.30T + 29T^{2} \)
31 \( 1 - 4.03T + 31T^{2} \)
41 \( 1 + (1.53 - 2.65i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 2.43T + 43T^{2} \)
47 \( 1 + 5.40T + 47T^{2} \)
53 \( 1 + (-3.91 - 6.77i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.83 + 6.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.56 + 2.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.614 + 1.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.69 - 2.93i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.76T + 73T^{2} \)
79 \( 1 + (-6.34 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.01 + 1.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.0T + 97T^{2} \)
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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

−9.119414757375985689628766872528, −8.195544272928135526710673500826, −7.30406160652861548828511115196, −6.51348094658998381087607636189, −5.87062056030465496083513753812, −4.83523892866618591543243806711, −4.22685974463000271996488250424, −3.19880897799181487032782769271, −1.97550276624280519252539574730, −1.03623725306129079246591329151, 0.853297674992731000922790301080, 2.19778040504948655645376661942, 3.05590644663491427302259916187, 4.04344091963660453837423640205, 4.88735476966402852858122524619, 5.88165903598750486991685849000, 6.65343891708031478354128232466, 7.08389630374652636247454285507, 8.167928728179624251217929255172, 8.846350460247597462101170076423

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