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Properties

Label 2-156-52.51-c2-0-9
Degree $2$
Conductor $156$
Sign $0.999 - 0.0259i$
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 − 0.731i)2-s + 1.73i·3-s + (2.92 + 2.72i)4-s − 3.15i·5-s + (1.26 − 3.22i)6-s + 3.05·7-s + (−3.46 − 7.21i)8-s − 2.99·9-s + (−2.30 + 5.87i)10-s + 3.10·11-s + (−4.71 + 5.07i)12-s + (9.09 + 9.28i)13-s + (−5.69 − 2.23i)14-s + 5.46·15-s + (1.16 + 15.9i)16-s + 20.0·17-s + ⋯
L(s)  = 1  + (−0.930 − 0.365i)2-s + 0.577i·3-s + (0.732 + 0.680i)4-s − 0.630i·5-s + (0.211 − 0.537i)6-s + 0.436·7-s + (−0.432 − 0.901i)8-s − 0.333·9-s + (−0.230 + 0.587i)10-s + 0.281·11-s + (−0.393 + 0.422i)12-s + (0.699 + 0.714i)13-s + (−0.406 − 0.159i)14-s + 0.364·15-s + (0.0728 + 0.997i)16-s + 1.18·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0259i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.999 - 0.0259i$
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: \((2,\ 156,\ (\ :1),\ 0.999 - 0.0259i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.06043 + 0.0137718i\)
\(L(\frac12)\) \(\approx\) \(1.06043 + 0.0137718i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.86 + 0.731i)T \)
3 \( 1 - 1.73iT \)
13 \( 1 + (-9.09 - 9.28i)T \)
good5 \( 1 + 3.15iT - 25T^{2} \)
7 \( 1 - 3.05T + 49T^{2} \)
11 \( 1 - 3.10T + 121T^{2} \)
17 \( 1 - 20.0T + 289T^{2} \)
19 \( 1 - 12.4T + 361T^{2} \)
23 \( 1 + 5.55iT - 529T^{2} \)
29 \( 1 - 20.2T + 841T^{2} \)
31 \( 1 - 7.89T + 961T^{2} \)
37 \( 1 - 43.9iT - 1.36e3T^{2} \)
41 \( 1 - 18.3iT - 1.68e3T^{2} \)
43 \( 1 + 43.4iT - 1.84e3T^{2} \)
47 \( 1 + 34.6T + 2.20e3T^{2} \)
53 \( 1 + 6.30T + 2.80e3T^{2} \)
59 \( 1 + 64.0T + 3.48e3T^{2} \)
61 \( 1 + 77.9T + 3.72e3T^{2} \)
67 \( 1 - 130.T + 4.48e3T^{2} \)
71 \( 1 + 80.1T + 5.04e3T^{2} \)
73 \( 1 + 112. iT - 5.32e3T^{2} \)
79 \( 1 + 98.8iT - 6.24e3T^{2} \)
83 \( 1 - 70.9T + 6.88e3T^{2} \)
89 \( 1 - 17.2iT - 7.92e3T^{2} \)
97 \( 1 + 27.5iT - 9.40e3T^{2} \)
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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

−12.28166701842461211148481291285, −11.61123692215119321746027174014, −10.57215115430186200535209283002, −9.593034730260213523740773155272, −8.742829409284072579941388149665, −7.87022730459749839645177399606, −6.38891862905906915554015353970, −4.78873722093158369362733259190, −3.32301545331919520283508577177, −1.30399722673782372034105321356, 1.21988545265205774467082789958, 3.02443385911778204227757130856, 5.44865132683654500042222717025, 6.51347280915394591464982253849, 7.57714200844470710893701845527, 8.318856512300200991499002809735, 9.594053670960905189706688373227, 10.66856647591207874772895895862, 11.44420742494753723495027503445, 12.52824805263962321699510534498

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