↑

Properties

Label 2-1148-41.9-c1-0-8
Degree $2$
Conductor $1148$
Sign $-0.478 - 0.878i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
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  + (1.62 + 1.62i)3-s + 3.63i·5-s + (0.707 + 0.707i)7-s + 2.29i·9-s + (1.60 + 1.60i)11-s + (2.82 + 2.82i)13-s + (−5.92 + 5.92i)15-s + (2.68 − 2.68i)17-s + (4.94 − 4.94i)19-s + 2.30i·21-s − 4.18·23-s − 8.24·25-s + (1.14 − 1.14i)27-s + (−1.01 − 1.01i)29-s − 6.11·31-s + ⋯
L(s)  = 1  + (0.939 + 0.939i)3-s + 1.62i·5-s + (0.267 + 0.267i)7-s + 0.766i·9-s + (0.485 + 0.485i)11-s + (0.784 + 0.784i)13-s + (−1.52 + 1.52i)15-s + (0.651 − 0.651i)17-s + (1.13 − 1.13i)19-s + 0.502i·21-s − 0.872·23-s − 1.64·25-s + (0.219 − 0.219i)27-s + (−0.188 − 0.188i)29-s − 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.478 - 0.878i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.478 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.465111347\)
\(L(\frac12)\) \(\approx\) \(2.465111347\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-5.19 - 3.74i)T \)
good3 \( 1 + (-1.62 - 1.62i)T + 3iT^{2} \)
5 \( 1 - 3.63iT - 5T^{2} \)
11 \( 1 + (-1.60 - 1.60i)T + 11iT^{2} \)
13 \( 1 + (-2.82 - 2.82i)T + 13iT^{2} \)
17 \( 1 + (-2.68 + 2.68i)T - 17iT^{2} \)
19 \( 1 + (-4.94 + 4.94i)T - 19iT^{2} \)
23 \( 1 + 4.18T + 23T^{2} \)
29 \( 1 + (1.01 + 1.01i)T + 29iT^{2} \)
31 \( 1 + 6.11T + 31T^{2} \)
37 \( 1 + 1.89T + 37T^{2} \)
43 \( 1 + 6.11iT - 43T^{2} \)
47 \( 1 + (1.11 - 1.11i)T - 47iT^{2} \)
53 \( 1 + (6.21 + 6.21i)T + 53iT^{2} \)
59 \( 1 + 2.32T + 59T^{2} \)
61 \( 1 + 10.9iT - 61T^{2} \)
67 \( 1 + (3.50 - 3.50i)T - 67iT^{2} \)
71 \( 1 + (-1.61 - 1.61i)T + 71iT^{2} \)
73 \( 1 + 3.53iT - 73T^{2} \)
79 \( 1 + (7.22 + 7.22i)T + 79iT^{2} \)
83 \( 1 - 0.636T + 83T^{2} \)
89 \( 1 + (2.14 + 2.14i)T + 89iT^{2} \)
97 \( 1 + (11.3 - 11.3i)T - 97iT^{2} \)
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

−9.732355086667502667112673066043, −9.555747511948924624771911178767, −8.594511246220907399428554692612, −7.53687797417961599847261556433, −6.91103879579123933233153245876, −5.92862650173815636542535365406, −4.66772875636286342053292329259, −3.65238610220274559224369393661, −3.09291815138758016658808072227, −2.03877767589737822798090391270, 1.10803805283838917182160726862, 1.62488702838545117040206490130, 3.28640068089882749224922320612, 4.09855985324904398049608595881, 5.43638003496752073010630039014, 6.00053500285870843329875051238, 7.49702024486237452259927102124, 7.973014622715772314097225555336, 8.555935945682099591704627578098, 9.203650131967607837040675291749

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