↑

Properties

Label 2-354-59.25-c1-0-6
Degree $2$
Conductor $354$
Sign $0.904 - 0.426i$
Analytic cond. $2.82670$
Root an. cond. $1.68128$
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  + (0.796 + 0.605i)2-s + (−0.468 − 0.883i)3-s + (0.267 + 0.963i)4-s + (2.16 + 2.05i)5-s + (0.161 − 0.986i)6-s + (2.55 − 3.00i)7-s + (−0.370 + 0.928i)8-s + (−0.561 + 0.827i)9-s + (0.482 + 2.94i)10-s + (−4.21 − 2.53i)11-s + (0.725 − 0.687i)12-s + (2.39 + 3.53i)13-s + (3.85 − 0.847i)14-s + (0.798 − 2.87i)15-s + (−0.856 + 0.515i)16-s + (2.93 + 3.45i)17-s + ⋯
L(s)  = 1  + (0.562 + 0.427i)2-s + (−0.270 − 0.510i)3-s + (0.133 + 0.481i)4-s + (0.968 + 0.917i)5-s + (0.0660 − 0.402i)6-s + (0.965 − 1.13i)7-s + (−0.130 + 0.328i)8-s + (−0.187 + 0.275i)9-s + (0.152 + 0.931i)10-s + (−1.26 − 0.763i)11-s + (0.209 − 0.198i)12-s + (0.665 + 0.980i)13-s + (1.02 − 0.226i)14-s + (0.206 − 0.742i)15-s + (−0.214 + 0.128i)16-s + (0.711 + 0.837i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.904 - 0.426i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ 0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95233 + 0.436750i\)
\(L(\frac12)\) \(\approx\) \(1.95233 + 0.436750i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.796 - 0.605i)T \)
3 \( 1 + (0.468 + 0.883i)T \)
59 \( 1 + (4.47 + 6.24i)T \)
good5 \( 1 + (-2.16 - 2.05i)T + (0.270 + 4.99i)T^{2} \)
7 \( 1 + (-2.55 + 3.00i)T + (-1.13 - 6.90i)T^{2} \)
11 \( 1 + (4.21 + 2.53i)T + (5.15 + 9.71i)T^{2} \)
13 \( 1 + (-2.39 - 3.53i)T + (-4.81 + 12.0i)T^{2} \)
17 \( 1 + (-2.93 - 3.45i)T + (-2.75 + 16.7i)T^{2} \)
19 \( 1 + (-2.82 + 1.30i)T + (12.3 - 14.4i)T^{2} \)
23 \( 1 + (5.01 + 1.68i)T + (18.3 + 13.9i)T^{2} \)
29 \( 1 + (-0.0463 + 0.0352i)T + (7.75 - 27.9i)T^{2} \)
31 \( 1 + (-5.76 - 2.66i)T + (20.0 + 23.6i)T^{2} \)
37 \( 1 + (4.00 + 10.0i)T + (-26.8 + 25.4i)T^{2} \)
41 \( 1 + (-1.68 + 0.568i)T + (32.6 - 24.8i)T^{2} \)
43 \( 1 + (10.0 - 6.03i)T + (20.1 - 37.9i)T^{2} \)
47 \( 1 + (8.86 - 8.40i)T + (2.54 - 46.9i)T^{2} \)
53 \( 1 + (-0.474 + 2.89i)T + (-50.2 - 16.9i)T^{2} \)
61 \( 1 + (10.6 + 8.05i)T + (16.3 + 58.7i)T^{2} \)
67 \( 1 + (-4.07 + 10.2i)T + (-48.6 - 46.0i)T^{2} \)
71 \( 1 + (7.94 - 7.52i)T + (3.84 - 70.8i)T^{2} \)
73 \( 1 + (0.904 - 0.199i)T + (66.2 - 30.6i)T^{2} \)
79 \( 1 + (3.19 - 6.02i)T + (-44.3 - 65.3i)T^{2} \)
83 \( 1 + (-2.24 + 0.244i)T + (81.0 - 17.8i)T^{2} \)
89 \( 1 + (0.491 - 0.373i)T + (23.8 - 85.7i)T^{2} \)
97 \( 1 + (-4.80 - 1.05i)T + (88.0 + 40.7i)T^{2} \)
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

−11.34503822324260989080094127714, −10.81758707403088486060964317145, −9.992188061149245153257208288036, −8.292774706826072191893617306000, −7.63664594368282892809721223629, −6.56920170914273406678866466480, −5.88377874863055873274633732225, −4.74129581992078187549462397372, −3.28765989399171777165325444229, −1.77662682994447489345648490914, 1.64518330979100493808869763613, 2.95828530318408272884170554562, 4.83236461732445677918561039806, 5.28354911689826233364219022433, 5.90816937168168613787088915179, 7.86001417175840232472577556659, 8.731696535491205687684339358109, 9.933962312297498287043546928590, 10.25998318616791904039099864368, 11.76093917421983583798176011348

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