Properties

Label 2-1216-19.18-c2-0-33
Degree $2$
Conductor $1216$
Sign $-0.526 - 0.850i$
Analytic cond. $33.1336$
Root an. cond. $5.75617$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.38i·3-s + 4·5-s + 7-s − 19.9·9-s + 14·11-s − 16.1i·13-s + 21.5i·15-s + 23·17-s + (10 + 16.1i)19-s + 5.38i·21-s + 23-s − 9·25-s − 59.2i·27-s + 48.4i·29-s + 32.3i·31-s + ⋯
L(s)  = 1  + 1.79i·3-s + 0.800·5-s + 0.142·7-s − 2.22·9-s + 1.27·11-s − 1.24i·13-s + 1.43i·15-s + 1.35·17-s + (0.526 + 0.850i)19-s + 0.256i·21-s + 0.0434·23-s − 0.359·25-s − 2.19i·27-s + 1.67i·29-s + 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.526 - 0.850i$
Analytic conductor: \(33.1336\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1),\ -0.526 - 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.470340094\)
\(L(\frac12)\) \(\approx\) \(2.470340094\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-10 - 16.1i)T \)
good3 \( 1 - 5.38iT - 9T^{2} \)
5 \( 1 - 4T + 25T^{2} \)
7 \( 1 - T + 49T^{2} \)
11 \( 1 - 14T + 121T^{2} \)
13 \( 1 + 16.1iT - 169T^{2} \)
17 \( 1 - 23T + 289T^{2} \)
23 \( 1 - T + 529T^{2} \)
29 \( 1 - 48.4iT - 841T^{2} \)
31 \( 1 - 32.3iT - 961T^{2} \)
37 \( 1 + 32.3iT - 1.36e3T^{2} \)
41 \( 1 - 32.3iT - 1.68e3T^{2} \)
43 \( 1 - 68T + 1.84e3T^{2} \)
47 \( 1 + 26T + 2.20e3T^{2} \)
53 \( 1 - 80.7iT - 2.80e3T^{2} \)
59 \( 1 - 16.1iT - 3.48e3T^{2} \)
61 \( 1 - 40T + 3.72e3T^{2} \)
67 \( 1 + 16.1iT - 4.48e3T^{2} \)
71 \( 1 - 32.3iT - 5.04e3T^{2} \)
73 \( 1 + 7T + 5.32e3T^{2} \)
79 \( 1 + 96.9iT - 6.24e3T^{2} \)
83 \( 1 - 32T + 6.88e3T^{2} \)
89 \( 1 + 129. iT - 7.92e3T^{2} \)
97 \( 1 + 96.9iT - 9.40e3T^{2} \)
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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

−9.849767427588763911175565021905, −9.260830292487217014082670146261, −8.522736283913833552330693976924, −7.47590301381111272033370804583, −5.98887171304563702153917503313, −5.57991657693264588891736896762, −4.71885420762624143477846122402, −3.61064707848301841295365696928, −3.08577152874536546932261570591, −1.30987901373179710315103427509, 0.818038274152329333782654862143, 1.70870053072043805899752136022, 2.48785799172814581395880074537, 3.90215098099036191626260203998, 5.35349135037206143523153771266, 6.24332934296524810959101456604, 6.65462470832823713973030884777, 7.54129496404459309833062785879, 8.249045386689032405343335985171, 9.322302678789353060981181437082

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