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Properties

Label 2-6e2-12.11-c5-0-8
Degree $2$
Conductor $36$
Sign $0.265 + 0.963i$
Analytic cond. $5.77381$
Root an. cond. $2.40287$
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  + (5.11 − 2.42i)2-s + (20.2 − 24.7i)4-s − 49.0i·5-s − 12.2i·7-s + (43.6 − 175. i)8-s + (−118. − 250. i)10-s + 706.·11-s − 795.·13-s + (−29.7 − 62.7i)14-s + (−201. − 1.00e3i)16-s + 363. i·17-s + 2.77e3i·19-s + (−1.21e3 − 994. i)20-s + (3.61e3 − 1.71e3i)22-s + 1.04e3·23-s + ⋯
L(s)  = 1  + (0.903 − 0.428i)2-s + (0.633 − 0.773i)4-s − 0.877i·5-s − 0.0946i·7-s + (0.241 − 0.970i)8-s + (−0.375 − 0.793i)10-s + 1.76·11-s − 1.30·13-s + (−0.0405 − 0.0855i)14-s + (−0.197 − 0.980i)16-s + 0.305i·17-s + 1.76i·19-s + (−0.679 − 0.556i)20-s + (1.59 − 0.753i)22-s + 0.413·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.265 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.265 + 0.963i$
Analytic conductor: \(5.77381\)
Root analytic conductor: \(2.40287\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :5/2),\ 0.265 + 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.07977 - 1.58367i\)
\(L(\frac12)\) \(\approx\) \(2.07977 - 1.58367i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.11 + 2.42i)T \)
3 \( 1 \)
good5 \( 1 + 49.0iT - 3.12e3T^{2} \)
7 \( 1 + 12.2iT - 1.68e4T^{2} \)
11 \( 1 - 706.T + 1.61e5T^{2} \)
13 \( 1 + 795.T + 3.71e5T^{2} \)
17 \( 1 - 363. iT - 1.41e6T^{2} \)
19 \( 1 - 2.77e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.04e3T + 6.43e6T^{2} \)
29 \( 1 - 4.96e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.93e3iT - 2.86e7T^{2} \)
37 \( 1 + 5.62e3T + 6.93e7T^{2} \)
41 \( 1 + 8.66e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.75e4iT - 1.47e8T^{2} \)
47 \( 1 - 97.2T + 2.29e8T^{2} \)
53 \( 1 + 2.18e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.92e4T + 7.14e8T^{2} \)
61 \( 1 - 931.T + 8.44e8T^{2} \)
67 \( 1 - 3.04e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.23e4T + 1.80e9T^{2} \)
73 \( 1 - 146.T + 2.07e9T^{2} \)
79 \( 1 - 3.78e4iT - 3.07e9T^{2} \)
83 \( 1 - 6.25e4T + 3.93e9T^{2} \)
89 \( 1 - 3.59e4iT - 5.58e9T^{2} \)
97 \( 1 + 9.63e4T + 8.58e9T^{2} \)
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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

−14.79084006275276118946888538804, −14.08461397540359106753861051766, −12.44763446893095081552462152890, −12.06891017309997381459341686750, −10.34460212550183929939814735851, −8.996306872666390480214538307305, −6.91159143048555505807249720750, −5.26828031076636835562792476185, −3.84072515797938757604058581394, −1.45716898570738438873144110731, 2.75597738701463492121005058277, 4.52321719955077363863882067909, 6.40980000845920004974007921441, 7.33544605460466892580202854946, 9.307335112480267200268946383099, 11.17988462891689355061128242873, 12.05828379563457518890005805000, 13.54863167486406692595366119734, 14.63347856049648886969128653611, 15.22612568486024090307147342725

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