↑

Properties

Label 2-296-37.10-c1-0-5
Degree $2$
Conductor $296$
Sign $0.825 + 0.564i$
Analytic cond. $2.36357$
Root an. cond. $1.53739$
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  + (−0.261 − 0.453i)3-s + (1.19 + 2.07i)5-s + (−2.55 − 4.43i)7-s + (1.36 − 2.36i)9-s + 5.32·11-s + (1.22 + 2.12i)13-s + (0.626 − 1.08i)15-s + (0.5 − 0.866i)17-s + (0.435 + 0.753i)19-s + (−1.33 + 2.32i)21-s + 5.98·23-s + (−0.364 + 0.631i)25-s − 2.99·27-s − 5.07·29-s − 5.32·31-s + ⋯
L(s)  = 1  + (−0.151 − 0.261i)3-s + (0.535 + 0.927i)5-s + (−0.967 − 1.67i)7-s + (0.454 − 0.786i)9-s + 1.60·11-s + (0.340 + 0.589i)13-s + (0.161 − 0.280i)15-s + (0.121 − 0.210i)17-s + (0.0998 + 0.172i)19-s + (−0.292 + 0.506i)21-s + 1.24·23-s + (−0.0729 + 0.126i)25-s − 0.576·27-s − 0.942·29-s − 0.956·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.825 + 0.564i$
Analytic conductor: \(2.36357\)
Root analytic conductor: \(1.53739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 296,\ (\ :1/2),\ 0.825 + 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28393 - 0.397174i\)
\(L(\frac12)\) \(\approx\) \(1.28393 - 0.397174i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (-1.34 + 5.93i)T \)
good3 \( 1 + (0.261 + 0.453i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.19 - 2.07i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.55 + 4.43i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.32T + 11T^{2} \)
13 \( 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.435 - 0.753i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.98T + 23T^{2} \)
29 \( 1 + 5.07T + 29T^{2} \)
31 \( 1 + 5.32T + 31T^{2} \)
41 \( 1 + (-3.52 - 6.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 1.39T + 43T^{2} \)
47 \( 1 + 12.6T + 47T^{2} \)
53 \( 1 + (-0.00720 + 0.0124i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.09 - 5.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.81 - 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.958 + 1.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.310 + 0.537i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.635T + 73T^{2} \)
79 \( 1 + (-7.73 - 13.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.691 - 1.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.52 + 9.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.20T + 97T^{2} \)
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

−11.50953334611138617225628191114, −10.76419346751654475128739292949, −9.706747690678274311744775317117, −9.257894204505877404814869000134, −7.27886885070470357297091868810, −6.79710451068509928979611814057, −6.19891550978095729714910976260, −4.10744856201714768422115724763, −3.38980016001362032063332799294, −1.23472202783376885707929203908, 1.75431081024579326265901795392, 3.40752441033752864148978242771, 4.99252304306547224653324108930, 5.73031527758087461453608356073, 6.73681081015423066891297789610, 8.353898921049164849073860298278, 9.302762080926139891346578753743, 9.518178616625585198562516343494, 10.99250001149921161911308441503, 11.97114813899950486163318110013

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