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Properties

Label 2-156-13.8-c2-0-0
Degree $2$
Conductor $156$
Sign $-0.0894 - 0.995i$
Analytic cond. $4.25069$
Root an. cond. $2.06172$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·3-s + (1.92 + 1.92i)5-s + (−3.58 + 3.58i)7-s + 2.99·9-s + (−6.83 + 6.83i)11-s + (4.86 + 12.0i)13-s + (−3.33 − 3.33i)15-s + 22.4i·17-s + (1.26 + 1.26i)19-s + (6.20 − 6.20i)21-s + 35.8i·23-s − 17.5i·25-s − 5.19·27-s + 5.29·29-s + (−22.8 − 22.8i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + (0.385 + 0.385i)5-s + (−0.512 + 0.512i)7-s + 0.333·9-s + (−0.621 + 0.621i)11-s + (0.374 + 0.927i)13-s + (−0.222 − 0.222i)15-s + 1.32i·17-s + (0.0663 + 0.0663i)19-s + (0.295 − 0.295i)21-s + 1.55i·23-s − 0.703i·25-s − 0.192·27-s + 0.182·29-s + (−0.738 − 0.738i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0894 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0894 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.0894 - 0.995i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ -0.0894 - 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.671786 + 0.734805i\)
\(L(\frac12)\) \(\approx\) \(0.671786 + 0.734805i\)
\(L(2)\) 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 \( 1 + 1.73T \)
13 \( 1 + (-4.86 - 12.0i)T \)
good5 \( 1 + (-1.92 - 1.92i)T + 25iT^{2} \)
7 \( 1 + (3.58 - 3.58i)T - 49iT^{2} \)
11 \( 1 + (6.83 - 6.83i)T - 121iT^{2} \)
17 \( 1 - 22.4iT - 289T^{2} \)
19 \( 1 + (-1.26 - 1.26i)T + 361iT^{2} \)
23 \( 1 - 35.8iT - 529T^{2} \)
29 \( 1 - 5.29T + 841T^{2} \)
31 \( 1 + (22.8 + 22.8i)T + 961iT^{2} \)
37 \( 1 + (-27.7 + 27.7i)T - 1.36e3iT^{2} \)
41 \( 1 + (2.72 + 2.72i)T + 1.68e3iT^{2} \)
43 \( 1 + 53.4iT - 1.84e3T^{2} \)
47 \( 1 + (-24.5 + 24.5i)T - 2.20e3iT^{2} \)
53 \( 1 - 20.8T + 2.80e3T^{2} \)
59 \( 1 + (38.5 - 38.5i)T - 3.48e3iT^{2} \)
61 \( 1 - 19.7T + 3.72e3T^{2} \)
67 \( 1 + (6.69 + 6.69i)T + 4.48e3iT^{2} \)
71 \( 1 + (-35.7 - 35.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (-68.0 + 68.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 3.68T + 6.24e3T^{2} \)
83 \( 1 + (40.8 + 40.8i)T + 6.88e3iT^{2} \)
89 \( 1 + (70.1 - 70.1i)T - 7.92e3iT^{2} \)
97 \( 1 + (-115. - 115. i)T + 9.40e3iT^{2} \)
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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

−12.87318909067843669607329916693, −12.00850800397021322925753175595, −10.92405501356852604103174718483, −10.01657304104119437333121839527, −9.052820215836070416878825533535, −7.57161461480262758111555798906, −6.38365671892080083310916245021, −5.54503062709514896125542361305, −3.93811725606744623950354369659, −2.08012702184040378854789811703, 0.67747638676501171582990430968, 3.05130794560738177349655490434, 4.80557303906700220403322897866, 5.83334997035610206887307519275, 7.00033794372138662100565235845, 8.268472039326140032335863338981, 9.528909671927670943892316877546, 10.48478988307388109744342461666, 11.31115447513076232755308381313, 12.65241843537883890222075814353

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