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Properties

Label 2-156-13.8-c2-0-1
Degree $2$
Conductor $156$
Sign $0.536 - 0.844i$
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 + (5.58 + 5.58i)5-s + (−7.42 + 7.42i)7-s + 2.99·9-s + (−6.03 + 6.03i)11-s + (3.49 − 12.5i)13-s + (9.66 + 9.66i)15-s − 12.4i·17-s + (26.2 + 26.2i)19-s + (−12.8 + 12.8i)21-s − 34.8i·23-s + 37.2i·25-s + 5.19·27-s + 23.0·29-s + (−1.49 − 1.49i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (1.11 + 1.11i)5-s + (−1.06 + 1.06i)7-s + 0.333·9-s + (−0.548 + 0.548i)11-s + (0.268 − 0.963i)13-s + (0.644 + 0.644i)15-s − 0.730i·17-s + (1.37 + 1.37i)19-s + (−0.612 + 0.612i)21-s − 1.51i·23-s + 1.49i·25-s + 0.192·27-s + 0.794·29-s + (−0.0480 − 0.0480i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.844i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.536 - 0.844i$
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: \((2,\ 156,\ (\ :1),\ 0.536 - 0.844i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.61334 + 0.886580i\)
\(L(\frac12)\) \(\approx\) \(1.61334 + 0.886580i\)
\(L(2)\) 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 \( 1 + (-3.49 + 12.5i)T \)
good5 \( 1 + (-5.58 - 5.58i)T + 25iT^{2} \)
7 \( 1 + (7.42 - 7.42i)T - 49iT^{2} \)
11 \( 1 + (6.03 - 6.03i)T - 121iT^{2} \)
17 \( 1 + 12.4iT - 289T^{2} \)
19 \( 1 + (-26.2 - 26.2i)T + 361iT^{2} \)
23 \( 1 + 34.8iT - 529T^{2} \)
29 \( 1 - 23.0T + 841T^{2} \)
31 \( 1 + (1.49 + 1.49i)T + 961iT^{2} \)
37 \( 1 + (-33.5 + 33.5i)T - 1.36e3iT^{2} \)
41 \( 1 + (51.0 + 51.0i)T + 1.68e3iT^{2} \)
43 \( 1 - 17.6iT - 1.84e3T^{2} \)
47 \( 1 + (-11.2 + 11.2i)T - 2.20e3iT^{2} \)
53 \( 1 - 17.2T + 2.80e3T^{2} \)
59 \( 1 + (16.2 - 16.2i)T - 3.48e3iT^{2} \)
61 \( 1 + 4.25T + 3.72e3T^{2} \)
67 \( 1 + (8.50 + 8.50i)T + 4.48e3iT^{2} \)
71 \( 1 + (96.2 + 96.2i)T + 5.04e3iT^{2} \)
73 \( 1 + (66.0 - 66.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 57.5T + 6.24e3T^{2} \)
83 \( 1 + (-39.7 - 39.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (5.18 - 5.18i)T - 7.92e3iT^{2} \)
97 \( 1 + (-112. - 112. i)T + 9.40e3iT^{2} \)
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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

−12.95290128609136274945205905996, −12.09187436349853018325584706368, −10.36141907290917849326703701281, −9.983721816785429867985776068070, −8.948279849110362861374677694318, −7.54573808164709290503346143349, −6.35634840488000617730601139863, −5.48238408798751432753977377803, −3.15985133199766105068840014317, −2.42918267419924887767220702086, 1.25065757863848383196200406871, 3.18791784779207540680097987522, 4.69572201147283387329559743568, 6.04762105906054551649575717250, 7.24117026771287229580129584555, 8.642999437050216121145209878341, 9.518678065633468970229942067221, 10.11887972868924379232847757692, 11.59247149906400154032389304218, 13.17028323588412162860685513343

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