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Properties

Label 2-198-11.3-c5-0-14
Degree $2$
Conductor $198$
Sign $0.981 + 0.191i$
Analytic cond. $31.7559$
Root an. cond. $5.63524$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.23 − 2.35i)2-s + (4.94 − 15.2i)4-s + (85.4 + 62.1i)5-s + (−3.56 + 10.9i)7-s + (−19.7 − 60.8i)8-s + 422.·10-s + (−58.1 − 397. i)11-s + (452. − 329. i)13-s + (14.2 + 43.8i)14-s + (−207. − 150. i)16-s + (55.4 + 40.2i)17-s + (690. + 2.12e3i)19-s + (1.36e3 − 993. i)20-s + (−1.12e3 − 1.14e3i)22-s + 2.60e3·23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (1.52 + 1.11i)5-s + (−0.0274 + 0.0845i)7-s + (−0.109 − 0.336i)8-s + 1.33·10-s + (−0.144 − 0.989i)11-s + (0.743 − 0.539i)13-s + (0.0194 + 0.0597i)14-s + (−0.202 − 0.146i)16-s + (0.0465 + 0.0338i)17-s + (0.438 + 1.34i)19-s + (0.764 − 0.555i)20-s + (−0.494 − 0.505i)22-s + 1.02·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(198\)    =    \(2 \cdot 3^{2} \cdot 11\)
Sign: $0.981 + 0.191i$
Analytic conductor: \(31.7559\)
Root analytic conductor: \(5.63524\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{198} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 198,\ (\ :5/2),\ 0.981 + 0.191i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.853916391\)
\(L(\frac12)\) \(\approx\) \(3.853916391\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.23 + 2.35i)T \)
3 \( 1 \)
11 \( 1 + (58.1 + 397. i)T \)
good5 \( 1 + (-85.4 - 62.1i)T + (965. + 2.97e3i)T^{2} \)
7 \( 1 + (3.56 - 10.9i)T + (-1.35e4 - 9.87e3i)T^{2} \)
13 \( 1 + (-452. + 329. i)T + (1.14e5 - 3.53e5i)T^{2} \)
17 \( 1 + (-55.4 - 40.2i)T + (4.38e5 + 1.35e6i)T^{2} \)
19 \( 1 + (-690. - 2.12e3i)T + (-2.00e6 + 1.45e6i)T^{2} \)
23 \( 1 - 2.60e3T + 6.43e6T^{2} \)
29 \( 1 + (-302. + 932. i)T + (-1.65e7 - 1.20e7i)T^{2} \)
31 \( 1 + (3.92e3 - 2.85e3i)T + (8.84e6 - 2.72e7i)T^{2} \)
37 \( 1 + (-3.14e3 + 9.68e3i)T + (-5.61e7 - 4.07e7i)T^{2} \)
41 \( 1 + (-3.08e3 - 9.48e3i)T + (-9.37e7 + 6.80e7i)T^{2} \)
43 \( 1 - 1.11e4T + 1.47e8T^{2} \)
47 \( 1 + (5.81e3 + 1.78e4i)T + (-1.85e8 + 1.34e8i)T^{2} \)
53 \( 1 + (-2.77e4 + 2.01e4i)T + (1.29e8 - 3.97e8i)T^{2} \)
59 \( 1 + (3.99e3 - 1.22e4i)T + (-5.78e8 - 4.20e8i)T^{2} \)
61 \( 1 + (6.90e3 + 5.01e3i)T + (2.60e8 + 8.03e8i)T^{2} \)
67 \( 1 + 3.34e4T + 1.35e9T^{2} \)
71 \( 1 + (-2.07e3 - 1.50e3i)T + (5.57e8 + 1.71e9i)T^{2} \)
73 \( 1 + (2.05e4 - 6.33e4i)T + (-1.67e9 - 1.21e9i)T^{2} \)
79 \( 1 + (5.65e4 - 4.10e4i)T + (9.50e8 - 2.92e9i)T^{2} \)
83 \( 1 + (7.70e4 + 5.59e4i)T + (1.21e9 + 3.74e9i)T^{2} \)
89 \( 1 - 3.34e4T + 5.58e9T^{2} \)
97 \( 1 + (-1.63e4 + 1.18e4i)T + (2.65e9 - 8.16e9i)T^{2} \)
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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

−11.33868470337233819766706521461, −10.64358681092045820697180674545, −9.937579728610593125257325237896, −8.790947913734164805774167579854, −7.17793338952514401389836960485, −5.91283420694913676174722290941, −5.61824462985616063138587268270, −3.54901811642621934873585282433, −2.64138638669013757810167823122, −1.30577872526733681955690676899, 1.20790988085806313585638460263, 2.49915975213790659184551506577, 4.43405248767914654504411553727, 5.23093759247817319612124398401, 6.23375458364320142278318814164, 7.31663302070085632104058938335, 8.894545160327186423008273448951, 9.342023897150150493342247404941, 10.59756573293331710944590302961, 11.91979864422317440828884285892

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