↑

Properties

Label 2-912-12.11-c1-0-10
Degree $2$
Conductor $912$
Sign $-0.880 - 0.474i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.909 + 1.47i)3-s + 2.62i·5-s + 3.58i·7-s + (−1.34 + 2.68i)9-s − 1.56·11-s + 1.23·13-s + (−3.87 + 2.39i)15-s − 3.38i·17-s − i·19-s + (−5.28 + 3.26i)21-s + 1.04·23-s − 1.91·25-s + (−5.17 + 0.456i)27-s − 7.20i·29-s + 1.42i·31-s + ⋯
L(s)  = 1  + (0.525 + 0.850i)3-s + 1.17i·5-s + 1.35i·7-s + (−0.448 + 0.893i)9-s − 0.472·11-s + 0.342·13-s + (−1.00 + 0.617i)15-s − 0.821i·17-s − 0.229i·19-s + (−1.15 + 0.711i)21-s + 0.217·23-s − 0.382·25-s + (−0.996 + 0.0878i)27-s − 1.33i·29-s + 0.256i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.880 - 0.474i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.880 - 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.403206 + 1.59819i\)
\(L(\frac12)\) \(\approx\) \(0.403206 + 1.59819i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.909 - 1.47i)T \)
19 \( 1 + iT \)
good5 \( 1 - 2.62iT - 5T^{2} \)
7 \( 1 - 3.58iT - 7T^{2} \)
11 \( 1 + 1.56T + 11T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 + 3.38iT - 17T^{2} \)
23 \( 1 - 1.04T + 23T^{2} \)
29 \( 1 + 7.20iT - 29T^{2} \)
31 \( 1 - 1.42iT - 31T^{2} \)
37 \( 1 - 4.55T + 37T^{2} \)
41 \( 1 + 10.8iT - 41T^{2} \)
43 \( 1 - 9.96iT - 43T^{2} \)
47 \( 1 - 0.136T + 47T^{2} \)
53 \( 1 - 9.69iT - 53T^{2} \)
59 \( 1 - 6.50T + 59T^{2} \)
61 \( 1 - 6.11T + 61T^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + 9.78T + 73T^{2} \)
79 \( 1 - 10.5iT - 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + 4.31iT - 89T^{2} \)
97 \( 1 - 6.67T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.35904975372395337629809019959, −9.615255980609368600106301775114, −8.883115383224758412114965596206, −8.069274723472033823150030103655, −7.11416824092498959462050749427, −5.98731714371967639069962720592, −5.25025893993626905013031412523, −4.06561337369795486122181921287, −2.79781380596455529385469343928, −2.50746330060335963041489707369, 0.74605771880589137388428055138, 1.71499295309222255617639417965, 3.33237689634001707726851186461, 4.25888442232235917714129299601, 5.33699912936106572756245956358, 6.47260161727910839463357695781, 7.30921392919844844350672263309, 8.138969915555853166517862142587, 8.648078837893142437660098593931, 9.633713719261608459274133406871

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