↑

Properties

Label 2-232-232.109-c1-0-2
Degree $2$
Conductor $232$
Sign $-0.649 - 0.760i$
Analytic cond. $1.85252$
Root an. cond. $1.36107$
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  + (−0.759 − 1.19i)2-s + (1.07 + 0.516i)3-s + (−0.845 + 1.81i)4-s + (−3.71 + 0.848i)5-s + (−0.198 − 1.67i)6-s + (−1.24 − 0.599i)7-s + (2.80 − 0.368i)8-s + (−0.986 − 1.23i)9-s + (3.83 + 3.78i)10-s + (−3.28 + 4.12i)11-s + (−1.84 + 1.50i)12-s + (−3.13 − 2.49i)13-s + (0.230 + 1.93i)14-s + (−4.42 − 1.01i)15-s + (−2.57 − 3.06i)16-s + 5.15i·17-s + ⋯
L(s)  = 1  + (−0.537 − 0.843i)2-s + (0.619 + 0.298i)3-s + (−0.422 + 0.906i)4-s + (−1.66 + 0.379i)5-s + (−0.0812 − 0.682i)6-s + (−0.470 − 0.226i)7-s + (0.991 − 0.130i)8-s + (−0.328 − 0.412i)9-s + (1.21 + 1.19i)10-s + (−0.991 + 1.24i)11-s + (−0.532 + 0.435i)12-s + (−0.868 − 0.692i)13-s + (0.0616 + 0.518i)14-s + (−1.14 − 0.260i)15-s + (−0.642 − 0.766i)16-s + 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $-0.649 - 0.760i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ -0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0663687 + 0.143918i\)
\(L(\frac12)\) \(\approx\) \(0.0663687 + 0.143918i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.759 + 1.19i)T \)
29 \( 1 + (1.22 - 5.24i)T \)
good3 \( 1 + (-1.07 - 0.516i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (3.71 - 0.848i)T + (4.50 - 2.16i)T^{2} \)
7 \( 1 + (1.24 + 0.599i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (3.28 - 4.12i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (3.13 + 2.49i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 - 5.15iT - 17T^{2} \)
19 \( 1 + (-0.763 + 0.367i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.615 + 2.69i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (1.58 - 0.362i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (-5.69 - 7.14i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 - 3.88iT - 41T^{2} \)
43 \( 1 + (-1.59 + 6.97i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (1.40 + 1.12i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.753 - 0.171i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 + (0.857 + 0.412i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (9.70 - 7.74i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (3.97 - 4.98i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (1.03 + 0.235i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (11.3 - 9.05i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (3.13 + 6.51i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (6.63 - 1.51i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-6.84 - 14.2i)T + (-60.4 + 75.8i)T^{2} \)
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

−12.48153964531216551836631902942, −11.54400835764533037501236289410, −10.47116675479839047764106885670, −9.871021592034501305593528705702, −8.558399370754176984867139216455, −7.86146923293606105561394405071, −7.05301651610187092323042717749, −4.65342471295861389508573026443, −3.61224316213417088177786171006, −2.74389367043731451499029949991, 0.13687265961568192441814396181, 2.91623578973949012568367921925, 4.53629117879782776419291163519, 5.71437641915245914894005839934, 7.44830376801626023430091871328, 7.66860412459567331123159620638, 8.677977249439097243727463004607, 9.434893216039674010339956569871, 10.96552590255559826743447725286, 11.68367054351949528762078318972

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