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Properties

Label 2-189-63.58-c3-0-5
Degree $2$
Conductor $189$
Sign $0.327 - 0.944i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.33·2-s − 6.21·4-s + (4.50 + 7.79i)5-s + (−3.16 − 18.2i)7-s + 18.9·8-s + (−6.01 − 10.4i)10-s + (14.6 − 25.4i)11-s + (−21.1 + 36.6i)13-s + (4.22 + 24.3i)14-s + 24.3·16-s + (2.56 + 4.45i)17-s + (−71.2 + 123. i)19-s + (−27.9 − 48.4i)20-s + (−19.6 + 33.9i)22-s + (89.0 + 154. i)23-s + ⋯
L(s)  = 1  − 0.472·2-s − 0.777·4-s + (0.402 + 0.697i)5-s + (−0.170 − 0.985i)7-s + 0.839·8-s + (−0.190 − 0.329i)10-s + (0.402 − 0.696i)11-s + (−0.451 + 0.782i)13-s + (0.0805 + 0.465i)14-s + 0.380·16-s + (0.0366 + 0.0634i)17-s + (−0.860 + 1.49i)19-s + (−0.312 − 0.541i)20-s + (−0.189 + 0.329i)22-s + (0.807 + 1.39i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.327 - 0.944i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.792582 + 0.564121i\)
\(L(\frac12)\) \(\approx\) \(0.792582 + 0.564121i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (3.16 + 18.2i)T \)
good2 \( 1 + 1.33T + 8T^{2} \)
5 \( 1 + (-4.50 - 7.79i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-14.6 + 25.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (21.1 - 36.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-2.56 - 4.45i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (71.2 - 123. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-89.0 - 154. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-109. - 189. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 147.T + 2.97e4T^{2} \)
37 \( 1 + (21.2 - 36.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (83.7 - 145. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-121. - 210. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 76.5T + 1.03e5T^{2} \)
53 \( 1 + (-181. - 314. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 121.T + 2.05e5T^{2} \)
61 \( 1 + 642.T + 2.26e5T^{2} \)
67 \( 1 + 162.T + 3.00e5T^{2} \)
71 \( 1 - 833.T + 3.57e5T^{2} \)
73 \( 1 + (62.4 + 108. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 842.T + 4.93e5T^{2} \)
83 \( 1 + (566. + 982. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-248. + 429. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (128. + 223. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.34173904701237237417225544851, −10.96529342665392019704173587629, −10.25325227086094587085258131732, −9.433990605840593961889904701032, −8.339469691668712603946940926684, −7.22845521547167843805660858405, −6.17181474100237876443966894765, −4.58808609335665889550863403043, −3.41764537781906773718863762508, −1.29523275373234575050591428424, 0.59127068816732825278880561412, 2.44705215915451892398160456280, 4.51520418574986116080638633475, 5.28233358696876737032575214638, 6.74127687878325058959359256568, 8.249826355324444027899801266502, 8.960715542171405496212820893265, 9.644579552030048023714576734204, 10.68407599913685903904125772217, 12.19631726269308324269985936226

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