↑

Properties

Label 2-296-296.101-c1-0-2
Degree $2$
Conductor $296$
Sign $0.908 - 0.418i$
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  + (−1.00 − 0.994i)2-s + (−2.16 + 1.24i)3-s + (0.0206 + 1.99i)4-s + (−0.890 − 1.54i)5-s + (3.41 + 0.896i)6-s + (−1.76 − 3.05i)7-s + (1.96 − 2.03i)8-s + (1.61 − 2.79i)9-s + (−0.639 + 2.43i)10-s + 4.13i·11-s + (−2.54 − 4.29i)12-s + (−0.0436 − 0.0756i)13-s + (−1.26 + 4.81i)14-s + (3.85 + 2.22i)15-s + (−3.99 + 0.0824i)16-s + (6.70 + 3.87i)17-s + ⋯
L(s)  = 1  + (−0.710 − 0.703i)2-s + (−1.24 + 0.720i)3-s + (0.0103 + 0.999i)4-s + (−0.398 − 0.689i)5-s + (1.39 + 0.365i)6-s + (−0.665 − 1.15i)7-s + (0.696 − 0.717i)8-s + (0.538 − 0.932i)9-s + (−0.202 + 0.770i)10-s + 1.24i·11-s + (−0.733 − 1.24i)12-s + (−0.0121 − 0.0209i)13-s + (−0.338 + 1.28i)14-s + (0.994 + 0.574i)15-s + (−0.999 + 0.0206i)16-s + (1.62 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.908 - 0.418i$
Analytic conductor: \(2.36357\)
Root analytic conductor: \(1.53739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 296,\ (\ :1/2),\ 0.908 - 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438502 + 0.0962400i\)
\(L(\frac12)\) \(\approx\) \(0.438502 + 0.0962400i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.00 + 0.994i)T \)
37 \( 1 + (-3.42 + 5.02i)T \)
good3 \( 1 + (2.16 - 1.24i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.890 + 1.54i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.76 + 3.05i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.13iT - 11T^{2} \)
13 \( 1 + (0.0436 + 0.0756i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.70 - 3.87i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.34 - 4.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.36iT - 23T^{2} \)
29 \( 1 + 3.98T + 29T^{2} \)
31 \( 1 - 1.77iT - 31T^{2} \)
41 \( 1 + (-4.07 - 7.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 5.20T + 43T^{2} \)
47 \( 1 + 4.78T + 47T^{2} \)
53 \( 1 + (-8.84 - 5.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.80 + 3.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.507 - 0.879i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.90 + 1.67i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.96 - 13.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 + (-3.90 + 2.25i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.49 + 4.32i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.29 + 4.21i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.43iT - 97T^{2} \)
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

−11.75513319593748603039745836968, −10.71403948182712390757755525852, −10.00286973998297247143019361562, −9.602681725557747198459262584136, −7.976355561499935584249827531065, −7.19643797466847152215149418948, −5.71417358908360539642526582064, −4.38328434160037812905414984202, −3.69158127480911962093431661275, −1.10285610436915826997111163068, 0.62525870046369957490939122784, 2.90739959048419881529054336403, 5.35958945115633516708427286219, 5.86753513295907513770551068790, 6.78758656392042722857818757675, 7.55992820108838619114371482691, 8.778863347856258080780461110472, 9.760937029227156605875762142438, 10.98067534102914517470817856774, 11.50120601402248711389732891146

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