↑

Properties

Label 1-19e2-361.188-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.692 + 0.721i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.870 + 0.492i)2-s + (−0.729 + 0.684i)3-s + (0.515 + 0.856i)4-s + (0.999 − 0.0367i)5-s + (−0.971 + 0.236i)6-s + (−0.191 + 0.981i)7-s + (0.0275 + 0.999i)8-s + (0.0642 − 0.997i)9-s + (0.888 + 0.459i)10-s + (−0.592 + 0.805i)11-s + (−0.962 − 0.272i)12-s + (0.919 − 0.393i)13-s + (−0.649 + 0.760i)14-s + (−0.703 + 0.710i)15-s + (−0.467 + 0.883i)16-s + (−0.999 − 0.0183i)17-s + ⋯
L(s)  = 1  + (0.870 + 0.492i)2-s + (−0.729 + 0.684i)3-s + (0.515 + 0.856i)4-s + (0.999 − 0.0367i)5-s + (−0.971 + 0.236i)6-s + (−0.191 + 0.981i)7-s + (0.0275 + 0.999i)8-s + (0.0642 − 0.997i)9-s + (0.888 + 0.459i)10-s + (−0.592 + 0.805i)11-s + (−0.962 − 0.272i)12-s + (0.919 − 0.393i)13-s + (−0.649 + 0.760i)14-s + (−0.703 + 0.710i)15-s + (−0.467 + 0.883i)16-s + (−0.999 − 0.0183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.692 + 0.721i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.692 + 0.721i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7332159773 + 1.721098644i\)
\(L(\frac12)\) \(\approx\) \(0.7332159773 + 1.721098644i\)
\(L(1)\) \(\approx\) \(1.153116382 + 1.001964165i\)
\(L(1)\) \(\approx\) \(1.153116382 + 1.001964165i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.870 + 0.492i)T \)
3 \( 1 + (-0.729 + 0.684i)T \)
5 \( 1 + (0.999 - 0.0367i)T \)
7 \( 1 + (-0.191 + 0.981i)T \)
11 \( 1 + (-0.592 + 0.805i)T \)
13 \( 1 + (0.919 - 0.393i)T \)
17 \( 1 + (-0.999 - 0.0183i)T \)
23 \( 1 + (0.957 - 0.289i)T \)
29 \( 1 + (-0.777 + 0.628i)T \)
31 \( 1 + (0.350 - 0.936i)T \)
37 \( 1 + (-0.401 - 0.915i)T \)
41 \( 1 + (-0.995 + 0.0917i)T \)
43 \( 1 + (-0.0459 + 0.998i)T \)
47 \( 1 + (0.690 + 0.723i)T \)
53 \( 1 + (0.280 - 0.959i)T \)
59 \( 1 + (0.418 - 0.908i)T \)
61 \( 1 + (-0.979 - 0.200i)T \)
67 \( 1 + (0.989 + 0.146i)T \)
71 \( 1 + (-0.979 + 0.200i)T \)
73 \( 1 + (0.484 + 0.875i)T \)
79 \( 1 + (-0.842 - 0.539i)T \)
83 \( 1 + (0.137 + 0.990i)T \)
89 \( 1 + (0.484 - 0.875i)T \)
97 \( 1 + (0.989 - 0.146i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.11113169420916146532922253815, −23.48137522556950773197439884499, −22.68990692424532943508837450255, −21.83915295221013979527708118908, −21.06113486577859501752466395019, −20.17903755258315801144080625552, −19.023733702884857713723617467240, −18.39580664581291819369984395365, −17.24710425640408167212694255266, −16.49329902726714224631183215059, −15.42186639217590889257075631790, −13.76647538009168424523053298834, −13.6222589887520216643448753334, −12.96523994984544829684369631111, −11.66582649872594681061777024063, −10.77394269660377838506594006924, −10.341254519781276660225317200232, −8.83406547930985363251646172633, −7.15771554108554477546653296876, −6.43920486929921571374097873246, −5.62357509577824828079795593579, −4.65000526848697210593587268081, −3.27094340968675963344064554507, −1.98347905682555938070973022324, −0.96665023582980521858608167654, 2.06022852122749678809409177879, 3.19236457675464282108501253406, 4.58766140785062317958473193637, 5.3757799147259631910198187244, 6.06512486012293352762980494317, 6.92420235389314264104897585097, 8.57586620883119810523821398809, 9.450101457135117407351832884729, 10.66338992092937800070999988730, 11.47437004634117453756221211223, 12.76860709957311593193215074152, 13.06753508528665222066101879654, 14.51552679901339405693613967083, 15.34465990993173233542485945927, 15.8956169757797658958661244295, 16.95500773283169364883253206740, 17.7581269306679186588594027671, 18.424810481389515668343864054869, 20.39612482954216005577223286349, 20.95007442594714244583743046355, 21.72069912159726662384486555038, 22.46081136032421898739445776559, 23.008874925472937184459203240451, 24.1450914252721722945788038540, 25.00875810255214791429239621216

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