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Properties

Label 2-189-63.5-c3-0-12
Degree $2$
Conductor $189$
Sign $0.729 + 0.684i$
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  − 1.15i·2-s + 6.67·4-s + (−0.137 − 0.237i)5-s + (18.5 + 0.122i)7-s − 16.9i·8-s + (−0.274 + 0.158i)10-s + (−22.7 − 13.1i)11-s + (39.6 + 22.8i)13-s + (0.141 − 21.3i)14-s + 33.8·16-s + (30.2 + 52.3i)17-s + (−38.8 − 22.4i)19-s + (−0.916 − 1.58i)20-s + (−15.1 + 26.1i)22-s + (110. − 63.8i)23-s + ⋯
L(s)  = 1  − 0.407i·2-s + 0.833·4-s + (−0.0122 − 0.0212i)5-s + (0.999 + 0.00663i)7-s − 0.747i·8-s + (−0.00867 + 0.00500i)10-s + (−0.622 − 0.359i)11-s + (0.845 + 0.487i)13-s + (0.00270 − 0.407i)14-s + 0.529·16-s + (0.431 + 0.747i)17-s + (−0.469 − 0.271i)19-s + (−0.0102 − 0.0177i)20-s + (−0.146 + 0.253i)22-s + (1.00 − 0.579i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.729 + 0.684i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.729 + 0.684i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.20465 - 0.872076i\)
\(L(\frac12)\) \(\approx\) \(2.20465 - 0.872076i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.5 - 0.122i)T \)
good2 \( 1 + 1.15iT - 8T^{2} \)
5 \( 1 + (0.137 + 0.237i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (22.7 + 13.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-39.6 - 22.8i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-30.2 - 52.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (38.8 + 22.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-110. + 63.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (58.2 - 33.6i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 98.9iT - 2.97e4T^{2} \)
37 \( 1 + (123. - 214. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-134. + 232. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (72.4 + 125. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 501.T + 1.03e5T^{2} \)
53 \( 1 + (333. - 192. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 548.T + 2.05e5T^{2} \)
61 \( 1 - 218. iT - 2.26e5T^{2} \)
67 \( 1 + 757.T + 3.00e5T^{2} \)
71 \( 1 - 1.10e3iT - 3.57e5T^{2} \)
73 \( 1 + (125. - 72.6i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 891.T + 4.93e5T^{2} \)
83 \( 1 + (-745. - 1.29e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (145. - 251. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (144. - 83.6i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.87781867994459814241081592156, −10.84436429206014010055057275864, −10.56459239144970754244933584607, −8.884496640731825347111026487217, −7.965127856148017437726723807678, −6.79532046800549284595126229038, −5.64734397053377674114176465525, −4.15554980819944994141765029815, −2.63974770686009783351857008072, −1.27675345949390932927427638027, 1.53749329269179209608285584168, 3.07310620236052834222213587378, 4.93462391157421913578571879237, 5.87013945012220270099552075185, 7.26448011366060320997604129846, 7.87545827694335370579331952002, 9.040478232679009560584284967553, 10.61357384792861365483659357922, 11.09312587018593209768900614261, 12.11907018842837081709649082414

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