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Properties

Label 2-912-76.75-c5-0-13
Degree $2$
Conductor $912$
Sign $-0.507 - 0.861i$
Analytic cond. $146.270$
Root an. cond. $12.0942$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s + 48.6·5-s − 194. i·7-s + 81·9-s + 630. i·11-s − 160. i·13-s + 438.·15-s − 1.13e3·17-s + (799. + 1.35e3i)19-s − 1.75e3i·21-s + 961. i·23-s − 754.·25-s + 729·27-s + 4.68e3i·29-s − 1.02e4·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.870·5-s − 1.50i·7-s + 0.333·9-s + 1.57i·11-s − 0.263i·13-s + 0.502·15-s − 0.951·17-s + (0.507 + 0.861i)19-s − 0.866i·21-s + 0.378i·23-s − 0.241·25-s + 0.192·27-s + 1.03i·29-s − 1.91·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.507 - 0.861i$
Analytic conductor: \(146.270\)
Root analytic conductor: \(12.0942\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :5/2),\ -0.507 - 0.861i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.420425706\)
\(L(\frac12)\) \(\approx\) \(1.420425706\)
\(L(\frac{7}{2})\) 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 \( 1 - 9T \)
19 \( 1 + (-799. - 1.35e3i)T \)
good5 \( 1 - 48.6T + 3.12e3T^{2} \)
7 \( 1 + 194. iT - 1.68e4T^{2} \)
11 \( 1 - 630. iT - 1.61e5T^{2} \)
13 \( 1 + 160. iT - 3.71e5T^{2} \)
17 \( 1 + 1.13e3T + 1.41e6T^{2} \)
23 \( 1 - 961. iT - 6.43e6T^{2} \)
29 \( 1 - 4.68e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.02e4T + 2.86e7T^{2} \)
37 \( 1 - 4.01e3iT - 6.93e7T^{2} \)
41 \( 1 - 6.40e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.47e4iT - 1.47e8T^{2} \)
47 \( 1 - 7.08e3iT - 2.29e8T^{2} \)
53 \( 1 - 803. iT - 4.18e8T^{2} \)
59 \( 1 + 4.46e4T + 7.14e8T^{2} \)
61 \( 1 + 3.14e4T + 8.44e8T^{2} \)
67 \( 1 + 5.65e4T + 1.35e9T^{2} \)
71 \( 1 - 3.01e4T + 1.80e9T^{2} \)
73 \( 1 + 3.18e4T + 2.07e9T^{2} \)
79 \( 1 - 7.82e4T + 3.07e9T^{2} \)
83 \( 1 - 3.83e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.40e5iT - 5.58e9T^{2} \)
97 \( 1 + 1.41e5iT - 8.58e9T^{2} \)
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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

−9.650121009885634821362525159785, −9.028209297189819343207331032862, −7.63467080465417890745842848537, −7.32808176600460941484584846048, −6.37149684368210992378371663593, −5.13931062269506583213826028196, −4.25532271781634760450063222772, −3.37972977050669415996776419146, −1.98681577667925266374803360541, −1.39788287893157540485514638963, 0.21637478495280036490055874587, 1.78363139765078483851714964119, 2.51110661342828641005199882672, 3.36470217018606370404258913802, 4.75572366748181826024515907669, 5.84369789744716856100899139703, 6.15812736646712576002045755973, 7.47006600206424063797585470096, 8.573871819489896586190014547324, 9.053215450109016246917077813716

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