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Properties

Label 2-108-36.11-c5-0-22
Degree $2$
Conductor $108$
Sign $0.0423 + 0.999i$
Analytic cond. $17.3214$
Root an. cond. $4.16190$
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  + (3.75 − 4.23i)2-s + (−3.81 − 31.7i)4-s + (52.1 − 30.1i)5-s + (185. + 106. i)7-s + (−148. − 103. i)8-s + (68.4 − 333. i)10-s + (185. − 321. i)11-s + (402. + 697. i)13-s + (1.14e3 − 382. i)14-s + (−994. + 242. i)16-s + 34.0i·17-s − 1.17e3i·19-s + (−1.15e3 − 1.54e3i)20-s + (−663. − 1.99e3i)22-s + (−1.98e3 − 3.43e3i)23-s + ⋯
L(s)  = 1  + (0.663 − 0.748i)2-s + (−0.119 − 0.992i)4-s + (0.933 − 0.538i)5-s + (1.42 + 0.825i)7-s + (−0.821 − 0.569i)8-s + (0.216 − 1.05i)10-s + (0.462 − 0.800i)11-s + (0.660 + 1.14i)13-s + (1.56 − 0.521i)14-s + (−0.971 + 0.236i)16-s + 0.0285i·17-s − 0.745i·19-s + (−0.646 − 0.862i)20-s + (−0.292 − 0.877i)22-s + (−0.780 − 1.35i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0423 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0423 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $0.0423 + 0.999i$
Analytic conductor: \(17.3214\)
Root analytic conductor: \(4.16190\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :5/2),\ 0.0423 + 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.51789 - 2.41334i\)
\(L(\frac12)\) \(\approx\) \(2.51789 - 2.41334i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.75 + 4.23i)T \)
3 \( 1 \)
good5 \( 1 + (-52.1 + 30.1i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-185. - 106. i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-185. + 321. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-402. - 697. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 34.0iT - 1.41e6T^{2} \)
19 \( 1 + 1.17e3iT - 2.47e6T^{2} \)
23 \( 1 + (1.98e3 + 3.43e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-382. - 221. i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-1.38e3 + 797. i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 1.10e4T + 6.93e7T^{2} \)
41 \( 1 + (-7.96e3 + 4.60e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (1.06e3 + 612. i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (1.22e3 - 2.11e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 2.02e4iT - 4.18e8T^{2} \)
59 \( 1 + (-1.70e3 - 2.95e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.99e4 - 3.45e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (6.08e3 - 3.51e3i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.56e4T + 1.80e9T^{2} \)
73 \( 1 - 6.54e4T + 2.07e9T^{2} \)
79 \( 1 + (-4.11e4 - 2.37e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (6.60e3 - 1.14e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 8.56e4iT - 5.58e9T^{2} \)
97 \( 1 + (2.79e4 - 4.84e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−12.37521038847896038470413584128, −11.55815877820994992302189177046, −10.70665965327824670956705226911, −9.193123948768478227742551230002, −8.616617996084987777584695169422, −6.33676656624115464813827424510, −5.37113390637262737487108572061, −4.29000512489572092894088066615, −2.28962382480565259506042171353, −1.28964451593344590034990464697, 1.76186742970093123781520220293, 3.67724830008799668944254512379, 5.03722612166750618138765511938, 6.12664070674670444460827381961, 7.40694406560428413587088291574, 8.240170192742410074302302199361, 9.876477880672448740730305330882, 10.96668930405764053668908875594, 12.13464963563731213235273715643, 13.43251939358864682823559979617

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