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Properties

Label 2-3192-133.132-c1-0-73
Degree $2$
Conductor $3192$
Sign $-0.293 + 0.955i$
Analytic cond. $25.4882$
Root an. cond. $5.04858$
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  + 3-s − 3.23i·5-s + (2.43 − 1.02i)7-s + 9-s − 0.249·11-s + 0.838·13-s − 3.23i·15-s − 7.10i·17-s + (3.33 + 2.80i)19-s + (2.43 − 1.02i)21-s − 8.88·23-s − 5.46·25-s + 27-s − 2.16i·29-s + 7.69·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.44i·5-s + (0.921 − 0.389i)7-s + 0.333·9-s − 0.0752·11-s + 0.232·13-s − 0.835i·15-s − 1.72i·17-s + (0.766 + 0.642i)19-s + (0.531 − 0.224i)21-s − 1.85·23-s − 1.09·25-s + 0.192·27-s − 0.401i·29-s + 1.38·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3192\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.293 + 0.955i$
Analytic conductor: \(25.4882\)
Root analytic conductor: \(5.04858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3192} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3192,\ (\ :1/2),\ -0.293 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.462713291\)
\(L(\frac12)\) \(\approx\) \(2.462713291\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + (-2.43 + 1.02i)T \)
19 \( 1 + (-3.33 - 2.80i)T \)
good5 \( 1 + 3.23iT - 5T^{2} \)
11 \( 1 + 0.249T + 11T^{2} \)
13 \( 1 - 0.838T + 13T^{2} \)
17 \( 1 + 7.10iT - 17T^{2} \)
23 \( 1 + 8.88T + 23T^{2} \)
29 \( 1 + 2.16iT - 29T^{2} \)
31 \( 1 - 7.69T + 31T^{2} \)
37 \( 1 + 7.96iT - 37T^{2} \)
41 \( 1 + 0.599T + 41T^{2} \)
43 \( 1 - 1.02T + 43T^{2} \)
47 \( 1 - 2.72iT - 47T^{2} \)
53 \( 1 - 8.82iT - 53T^{2} \)
59 \( 1 + 6.38T + 59T^{2} \)
61 \( 1 - 1.83iT - 61T^{2} \)
67 \( 1 - 6.12iT - 67T^{2} \)
71 \( 1 + 2.14iT - 71T^{2} \)
73 \( 1 - 1.16iT - 73T^{2} \)
79 \( 1 - 1.24iT - 79T^{2} \)
83 \( 1 - 5.96iT - 83T^{2} \)
89 \( 1 + 0.399T + 89T^{2} \)
97 \( 1 - 16.7T + 97T^{2} \)
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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

−8.363175182886695647604673218158, −7.85299966660333437936892289934, −7.31081222517728894802365641949, −6.00139679820184155944225078147, −5.24093009077481089805945399537, −4.51000545666460916599395286560, −3.96278434387660677770291995684, −2.66045791197878460487413919144, −1.60161274253188966063158984004, −0.71117224213667830809525679592, 1.58772408811687734743049907459, 2.39693985803590391897994241421, 3.26618709340973295325455315506, 4.03082980758852542607294392614, 5.01911781435135400327200285563, 6.13687989144050612573224746219, 6.54211068747200472264741774272, 7.60315669249424745658962975544, 8.083773380597704350757657710586, 8.667716892479094613153175263237

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