↑

Properties

Label 2-370-185.174-c1-0-9
Degree $2$
Conductor $370$
Sign $0.478 - 0.877i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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  + (−0.866 + 0.5i)2-s + (2.73 + 1.57i)3-s + (0.499 − 0.866i)4-s + (2.12 − 0.710i)5-s − 3.15·6-s + (1.45 + 0.837i)7-s + 0.999i·8-s + (3.48 + 6.02i)9-s + (−1.48 + 1.67i)10-s − 4.48·11-s + (2.73 − 1.57i)12-s + (−4.18 − 2.41i)13-s − 1.67·14-s + (6.91 + 1.40i)15-s + (−0.5 − 0.866i)16-s + (2.14 − 1.24i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (1.57 + 0.911i)3-s + (0.249 − 0.433i)4-s + (0.948 − 0.317i)5-s − 1.28·6-s + (0.548 + 0.316i)7-s + 0.353i·8-s + (1.16 + 2.00i)9-s + (−0.468 + 0.529i)10-s − 1.35·11-s + (0.789 − 0.455i)12-s + (−1.16 − 0.670i)13-s − 0.447·14-s + (1.78 + 0.362i)15-s + (−0.125 − 0.216i)16-s + (0.521 − 0.300i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.478 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.478 - 0.877i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.478 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60362 + 0.951837i\)
\(L(\frac12)\) \(\approx\) \(1.60362 + 0.951837i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-2.12 + 0.710i)T \)
37 \( 1 + (0.920 - 6.01i)T \)
good3 \( 1 + (-2.73 - 1.57i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.45 - 0.837i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.48T + 11T^{2} \)
13 \( 1 + (4.18 + 2.41i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.14 + 1.24i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.28 + 3.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.67iT - 23T^{2} \)
29 \( 1 + 9.08T + 29T^{2} \)
31 \( 1 + 2.83T + 31T^{2} \)
41 \( 1 + (-0.175 + 0.303i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 3.80iT - 43T^{2} \)
47 \( 1 - 2.63iT - 47T^{2} \)
53 \( 1 + (4.04 - 2.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.91 - 8.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.83 + 11.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.8 - 6.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.80 - 4.86i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.13iT - 73T^{2} \)
79 \( 1 + (-5.04 + 8.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.80 - 3.35i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.27 + 9.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.18iT - 97T^{2} \)
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

−11.08816301408049003083488650388, −10.06516341781305098734677846668, −9.711104683367017787207771154481, −8.849262113126446363278623906446, −8.025112018732295878896944765128, −7.35459688916281260337386602569, −5.36931971963303637773935260323, −4.89627095428034964698152568981, −2.95198211769203215773749548524, −2.14976213153043437030330361612, 1.72906535120103652237950128029, 2.39826408040818584451979455558, 3.59750241567011357018194586719, 5.49397671810264956188418040361, 7.11447580457608144350367135964, 7.56461923983112337218400851705, 8.386956063001997020661319054282, 9.504688333717663172875738595905, 9.918771821237743735431036460068, 11.08138212521851374740666509405

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