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Properties

Label 2-156-52.51-c2-0-18
Degree $2$
Conductor $156$
Sign $0.967 + 0.254i$
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  + (1.86 + 0.715i)2-s − 1.73i·3-s + (2.97 + 2.67i)4-s − 4.65i·5-s + (1.23 − 3.23i)6-s + 2.30·7-s + (3.64 + 7.12i)8-s − 2.99·9-s + (3.33 − 8.70i)10-s + 18.8·11-s + (4.62 − 5.15i)12-s + (−5.93 − 11.5i)13-s + (4.29 + 1.64i)14-s − 8.06·15-s + (1.71 + 15.9i)16-s − 7.23·17-s + ⋯
L(s)  = 1  + (0.933 + 0.357i)2-s − 0.577i·3-s + (0.744 + 0.668i)4-s − 0.931i·5-s + (0.206 − 0.539i)6-s + 0.328·7-s + (0.455 + 0.890i)8-s − 0.333·9-s + (0.333 − 0.870i)10-s + 1.71·11-s + (0.385 − 0.429i)12-s + (−0.456 − 0.889i)13-s + (0.307 + 0.117i)14-s − 0.537·15-s + (0.107 + 0.994i)16-s − 0.425·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.967 + 0.254i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ 0.967 + 0.254i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.55843 - 0.330908i\)
\(L(\frac12)\) \(\approx\) \(2.55843 - 0.330908i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.86 - 0.715i)T \)
3 \( 1 + 1.73iT \)
13 \( 1 + (5.93 + 11.5i)T \)
good5 \( 1 + 4.65iT - 25T^{2} \)
7 \( 1 - 2.30T + 49T^{2} \)
11 \( 1 - 18.8T + 121T^{2} \)
17 \( 1 + 7.23T + 289T^{2} \)
19 \( 1 + 17.0T + 361T^{2} \)
23 \( 1 - 43.7iT - 529T^{2} \)
29 \( 1 + 52.6T + 841T^{2} \)
31 \( 1 - 10.5T + 961T^{2} \)
37 \( 1 + 12.2iT - 1.36e3T^{2} \)
41 \( 1 - 38.5iT - 1.68e3T^{2} \)
43 \( 1 + 12.0iT - 1.84e3T^{2} \)
47 \( 1 + 62.7T + 2.20e3T^{2} \)
53 \( 1 - 74.8T + 2.80e3T^{2} \)
59 \( 1 - 11.4T + 3.48e3T^{2} \)
61 \( 1 + 32.1T + 3.72e3T^{2} \)
67 \( 1 - 60.8T + 4.48e3T^{2} \)
71 \( 1 - 8.07T + 5.04e3T^{2} \)
73 \( 1 + 58.3iT - 5.32e3T^{2} \)
79 \( 1 + 59.0iT - 6.24e3T^{2} \)
83 \( 1 - 48.7T + 6.88e3T^{2} \)
89 \( 1 - 80.5iT - 7.92e3T^{2} \)
97 \( 1 - 15.3iT - 9.40e3T^{2} \)
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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

−12.83829320910624889455753950847, −11.91066239964855858784347704328, −11.20223559117478738164669712998, −9.343745260109735153304978648442, −8.278655365143197884345507773259, −7.23345171931181613842979104078, −6.07320614910834578579712820539, −4.97352309002645073464463187805, −3.68728156623746298880729142336, −1.67358854898600830101095351710, 2.18153816164259221309868908292, 3.75920152179247632423710123864, 4.63051754395129837937915281177, 6.31615245109954222720084561279, 6.92786730495726104966867252546, 8.864686565484412017160674416962, 9.989765077679961666192724991800, 11.02642404123765320837620823204, 11.57697223707726242144308989488, 12.66033921204798552146903621487

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