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Properties

Label 2-189-63.38-c3-0-5
Degree $2$
Conductor $189$
Sign $0.278 + 0.960i$
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  + 5.00i·2-s − 17.0·4-s + (−9.28 + 16.0i)5-s + (12.9 + 13.2i)7-s − 45.5i·8-s + (−80.5 − 46.5i)10-s + (−30.4 + 17.5i)11-s + (−3.94 + 2.27i)13-s + (−66.2 + 65.0i)14-s + 91.5·16-s + (20.7 − 36.0i)17-s + (64.8 − 37.4i)19-s + (158. − 274. i)20-s + (−88.0 − 152. i)22-s + (132. + 76.3i)23-s + ⋯
L(s)  = 1  + 1.77i·2-s − 2.13·4-s + (−0.830 + 1.43i)5-s + (0.700 + 0.713i)7-s − 2.01i·8-s + (−2.54 − 1.47i)10-s + (−0.834 + 0.481i)11-s + (−0.0841 + 0.0485i)13-s + (−1.26 + 1.24i)14-s + 1.43·16-s + (0.296 − 0.513i)17-s + (0.783 − 0.452i)19-s + (1.77 − 3.07i)20-s + (−0.852 − 1.47i)22-s + (1.19 + 0.692i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.278 + 0.960i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.278 + 0.960i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.690593 - 0.518895i\)
\(L(\frac12)\) \(\approx\) \(0.690593 - 0.518895i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.9 - 13.2i)T \)
good2 \( 1 - 5.00iT - 8T^{2} \)
5 \( 1 + (9.28 - 16.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (30.4 - 17.5i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (3.94 - 2.27i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-20.7 + 36.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-64.8 + 37.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-132. - 76.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (42.5 + 24.5i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 101. iT - 2.97e4T^{2} \)
37 \( 1 + (-2.86 - 4.96i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (6.35 + 10.9i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-69.1 + 119. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 538.T + 1.03e5T^{2} \)
53 \( 1 + (-105. - 60.8i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 312.T + 2.05e5T^{2} \)
61 \( 1 - 504. iT - 2.26e5T^{2} \)
67 \( 1 + 654.T + 3.00e5T^{2} \)
71 \( 1 - 891. iT - 3.57e5T^{2} \)
73 \( 1 + (315. + 182. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 485.T + 4.93e5T^{2} \)
83 \( 1 + (248. - 430. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (685. + 1.18e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-205. - 118. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.30669160666585962362602786790, −11.87439447518731800567592266453, −10.93862899988608752426572962174, −9.602782802544947248678707919164, −8.404637035240022862030761007325, −7.43376176594248419120198519258, −7.06964072189788454942486186645, −5.66436399283074730293744519189, −4.71604146847268419362452679182, −3.01200567556811952079185615660, 0.41922036640279300758646972920, 1.44047440269509257227597205197, 3.32014179784221352850045122966, 4.44802027928958449886607765147, 5.18299316346960258424053415251, 7.79359934448869297720309545857, 8.476482407082330654414892055960, 9.524812115355781361361430338403, 10.65799217624004600180994548405, 11.34587893172244102607835516133

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