↑

Properties

Label 2-352-88.59-c2-0-2
Degree $2$
Conductor $352$
Sign $-0.538 + 0.842i$
Analytic cond. $9.59130$
Root an. cond. $3.09698$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.58 + 4.89i)3-s + (2.21 + 3.05i)5-s + (−8.10 + 2.63i)7-s + (−14.1 − 10.2i)9-s + (1.57 − 10.8i)11-s + (−10.4 + 14.4i)13-s + (−18.4 + 6.00i)15-s + (10.2 − 7.44i)17-s + (−5.03 + 15.4i)19-s − 43.8i·21-s − 2.47i·23-s + (3.32 − 10.2i)25-s + (35.2 − 25.5i)27-s + (24.8 − 8.08i)29-s + (8.88 − 12.2i)31-s + ⋯
L(s)  = 1  + (−0.529 + 1.63i)3-s + (0.443 + 0.610i)5-s + (−1.15 + 0.376i)7-s + (−1.56 − 1.14i)9-s + (0.143 − 0.989i)11-s + (−0.807 + 1.11i)13-s + (−1.23 + 0.400i)15-s + (0.602 − 0.437i)17-s + (−0.264 + 0.815i)19-s − 2.08i·21-s − 0.107i·23-s + (0.132 − 0.408i)25-s + (1.30 − 0.948i)27-s + (0.858 − 0.278i)29-s + (0.286 − 0.394i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $-0.538 + 0.842i$
Analytic conductor: \(9.59130\)
Root analytic conductor: \(3.09698\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1),\ -0.538 + 0.842i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.226503 - 0.413589i\)
\(L(\frac12)\) \(\approx\) \(0.226503 - 0.413589i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-1.57 + 10.8i)T \)
good3 \( 1 + (1.58 - 4.89i)T + (-7.28 - 5.29i)T^{2} \)
5 \( 1 + (-2.21 - 3.05i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (8.10 - 2.63i)T + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (10.4 - 14.4i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (-10.2 + 7.44i)T + (89.3 - 274. i)T^{2} \)
19 \( 1 + (5.03 - 15.4i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + 2.47iT - 529T^{2} \)
29 \( 1 + (-24.8 + 8.08i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-8.88 + 12.2i)T + (-296. - 913. i)T^{2} \)
37 \( 1 + (67.8 - 22.0i)T + (1.10e3 - 804. i)T^{2} \)
41 \( 1 + (2.18 - 6.73i)T + (-1.35e3 - 988. i)T^{2} \)
43 \( 1 - 12.3T + 1.84e3T^{2} \)
47 \( 1 + (23.6 + 7.67i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (31.4 - 43.2i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (32.0 + 98.5i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (26.4 + 36.3i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 - 16.3T + 4.48e3T^{2} \)
71 \( 1 + (-14.0 - 19.3i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-21.0 - 64.9i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (84.8 - 116. i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (78.9 - 57.3i)T + (2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 15.4T + 7.92e3T^{2} \)
97 \( 1 + (36.5 + 26.5i)T + (2.90e3 + 8.94e3i)T^{2} \)
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

−11.68036293369336563192597883533, −10.74172134811512615320124267983, −9.890967027154564560193041422384, −9.577893588170168632744322654715, −8.497964662725336449314961604922, −6.66154034039174186187075409687, −6.03332487443105919586320450134, −4.96892351130156895732312067405, −3.74121863851573781586889438437, −2.82325198277218906731383210300, 0.22522449542326376946816712268, 1.55865211970770939853861109396, 2.97255668771009832844556301484, 4.96418584398445125670280950861, 5.92383818112449590537781750779, 6.92062939612316145842659854644, 7.44337762487774730188483063690, 8.657528119982783984733267959148, 9.811213093510174513919938732158, 10.62326692190469996388294449369

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