Properties

Label 2-189-63.58-c3-0-15
Degree $2$
Conductor $189$
Sign $0.877 - 0.479i$
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  + 5.09·2-s + 17.9·4-s + (3.42 + 5.92i)5-s + (−0.241 + 18.5i)7-s + 50.6·8-s + (17.4 + 30.1i)10-s + (−20.5 + 35.5i)11-s + (31.8 − 55.2i)13-s + (−1.23 + 94.3i)14-s + 114.·16-s + (−38.0 − 65.8i)17-s + (46.7 − 81.0i)19-s + (61.3 + 106. i)20-s + (−104. + 181. i)22-s + (21.0 + 36.4i)23-s + ⋯
L(s)  = 1  + 1.80·2-s + 2.24·4-s + (0.305 + 0.529i)5-s + (−0.0130 + 0.999i)7-s + 2.24·8-s + (0.550 + 0.954i)10-s + (−0.562 + 0.974i)11-s + (0.680 − 1.17i)13-s + (−0.0234 + 1.80i)14-s + 1.79·16-s + (−0.542 − 0.939i)17-s + (0.564 − 0.978i)19-s + (0.686 + 1.18i)20-s + (−1.01 + 1.75i)22-s + (0.190 + 0.330i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.877 - 0.479i$
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: \((2,\ 189,\ (\ :3/2),\ 0.877 - 0.479i)\)

Particular Values

\(L(2)\) \(\approx\) \(4.88692 + 1.24835i\)
\(L(\frac12)\) \(\approx\) \(4.88692 + 1.24835i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.241 - 18.5i)T \)
good2 \( 1 - 5.09T + 8T^{2} \)
5 \( 1 + (-3.42 - 5.92i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (20.5 - 35.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-31.8 + 55.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (38.0 + 65.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-46.7 + 81.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-21.0 - 36.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (21.9 + 37.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 200.T + 2.97e4T^{2} \)
37 \( 1 + (52.1 - 90.3i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-108. + 188. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-236. - 409. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 17.8T + 1.03e5T^{2} \)
53 \( 1 + (211. + 365. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 291.T + 2.05e5T^{2} \)
61 \( 1 + 154.T + 2.26e5T^{2} \)
67 \( 1 + 838.T + 3.00e5T^{2} \)
71 \( 1 - 940.T + 3.57e5T^{2} \)
73 \( 1 + (-65.8 - 113. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 620.T + 4.93e5T^{2} \)
83 \( 1 + (102. + 177. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (432. - 749. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-331. - 573. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.52282040108092282006926532795, −11.46088270966164813424381110210, −10.70383670918535648950508938229, −9.335207560312655380622661146569, −7.64247459566386838267727136322, −6.59192894069885667126404245223, −5.55968724840907553698978961547, −4.79268285813109447715877332517, −3.13651030359737801051400587292, −2.34143396445639404175474095653, 1.62613967323751785415608664552, 3.45236768278967973249431626839, 4.28983844841427263367113609425, 5.51267386225471606336675450968, 6.37890263511291994288981795474, 7.53641973531072030850228516746, 8.971501573674485677886350886744, 10.66696237190502790747993997042, 11.16700003328824526477828672377, 12.41596316496579446139113188664

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