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Properties

Label 2-156-13.10-c3-0-4
Degree $2$
Conductor $156$
Sign $-0.923 + 0.383i$
Analytic cond. $9.20429$
Root an. cond. $3.03385$
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  + (1.5 + 2.59i)3-s − 11.8i·5-s + (−19.2 − 11.1i)7-s + (−4.5 + 7.79i)9-s + (−49.0 + 28.3i)11-s + (−28.0 + 37.5i)13-s + (30.9 − 17.8i)15-s + (20.5 − 35.6i)17-s + (−99.4 − 57.4i)19-s − 66.6i·21-s + (−33.8 − 58.6i)23-s − 16.5·25-s − 27·27-s + (67.1 + 116. i)29-s − 265. i·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 1.06i·5-s + (−1.03 − 0.600i)7-s + (−0.166 + 0.288i)9-s + (−1.34 + 0.776i)11-s + (−0.598 + 0.801i)13-s + (0.532 − 0.307i)15-s + (0.293 − 0.508i)17-s + (−1.20 − 0.693i)19-s − 0.692i·21-s + (−0.306 − 0.531i)23-s − 0.132·25-s − 0.192·27-s + (0.429 + 0.744i)29-s − 1.54i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.923 + 0.383i$
Analytic conductor: \(9.20429\)
Root analytic conductor: \(3.03385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :3/2),\ -0.923 + 0.383i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0681430 - 0.342227i\)
\(L(\frac12)\) \(\approx\) \(0.0681430 - 0.342227i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 - 2.59i)T \)
13 \( 1 + (28.0 - 37.5i)T \)
good5 \( 1 + 11.8iT - 125T^{2} \)
7 \( 1 + (19.2 + 11.1i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (49.0 - 28.3i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-20.5 + 35.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (99.4 + 57.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (33.8 + 58.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-67.1 - 116. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 265. iT - 2.97e4T^{2} \)
37 \( 1 + (60.0 - 34.6i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (99.0 - 57.2i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-153. + 266. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 518. iT - 1.03e5T^{2} \)
53 \( 1 - 304.T + 1.48e5T^{2} \)
59 \( 1 + (608. + 351. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-324. + 562. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (622. - 359. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-782. - 451. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 562. iT - 3.89e5T^{2} \)
79 \( 1 - 530.T + 4.93e5T^{2} \)
83 \( 1 - 986. iT - 5.71e5T^{2} \)
89 \( 1 + (271. - 156. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (120. + 69.5i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.36545759549883287782270115675, −10.79583597529669144567126954787, −9.854060851407813824004675382193, −9.139441971227137319028454329180, −7.929024435895338178212357791152, −6.73352810428569614319637625662, −5.07508647671346409852654154993, −4.23762912723302830297104910332, −2.50438578849629805766591237598, −0.14625104279669167634750769432, 2.52800787403769465469585627819, 3.32701636147664394754716206633, 5.61152269980539892657256239224, 6.49068326972838892968519756046, 7.67533212232075043105879321638, 8.623443492291399811647328070140, 10.14288655224837100589656174505, 10.63961858868116878290135619134, 12.15540750039775824348842829950, 12.88640963354841638479906791062

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