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Properties

Label 2-156-4.3-c2-0-13
Degree $2$
Conductor $156$
Sign $0.900 - 0.435i$
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  + (1.69 + 1.06i)2-s − 1.73i·3-s + (1.74 + 3.60i)4-s + 5.51·5-s + (1.84 − 2.93i)6-s − 1.01i·7-s + (−0.880 + 7.95i)8-s − 2.99·9-s + (9.34 + 5.86i)10-s − 5.63i·11-s + (6.23 − 3.01i)12-s + 3.60·13-s + (1.07 − 1.71i)14-s − 9.55i·15-s + (−9.94 + 12.5i)16-s + 8.22·17-s + ⋯
L(s)  = 1  + (0.847 + 0.531i)2-s − 0.577i·3-s + (0.435 + 0.900i)4-s + 1.10·5-s + (0.306 − 0.489i)6-s − 0.144i·7-s + (−0.110 + 0.993i)8-s − 0.333·9-s + (0.934 + 0.586i)10-s − 0.511i·11-s + (0.519 − 0.251i)12-s + 0.277·13-s + (0.0768 − 0.122i)14-s − 0.637i·15-s + (−0.621 + 0.783i)16-s + 0.483·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.900 - 0.435i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ 0.900 - 0.435i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.54769 + 0.583160i\)
\(L(\frac12)\) \(\approx\) \(2.54769 + 0.583160i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.69 - 1.06i)T \)
3 \( 1 + 1.73iT \)
13 \( 1 - 3.60T \)
good5 \( 1 - 5.51T + 25T^{2} \)
7 \( 1 + 1.01iT - 49T^{2} \)
11 \( 1 + 5.63iT - 121T^{2} \)
17 \( 1 - 8.22T + 289T^{2} \)
19 \( 1 - 11.5iT - 361T^{2} \)
23 \( 1 - 5.25iT - 529T^{2} \)
29 \( 1 + 48.7T + 841T^{2} \)
31 \( 1 + 59.3iT - 961T^{2} \)
37 \( 1 + 19.4T + 1.36e3T^{2} \)
41 \( 1 + 65.6T + 1.68e3T^{2} \)
43 \( 1 + 14.0iT - 1.84e3T^{2} \)
47 \( 1 - 9.34iT - 2.20e3T^{2} \)
53 \( 1 + 25.9T + 2.80e3T^{2} \)
59 \( 1 - 38.4iT - 3.48e3T^{2} \)
61 \( 1 - 72.1T + 3.72e3T^{2} \)
67 \( 1 + 101. iT - 4.48e3T^{2} \)
71 \( 1 - 80.3iT - 5.04e3T^{2} \)
73 \( 1 - 5.60T + 5.32e3T^{2} \)
79 \( 1 - 28.5iT - 6.24e3T^{2} \)
83 \( 1 - 103. iT - 6.88e3T^{2} \)
89 \( 1 - 69.8T + 7.92e3T^{2} \)
97 \( 1 + 27.6T + 9.40e3T^{2} \)
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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

−13.17037895761826898551064083841, −12.04914336624146226243002876473, −11.03734727028338215177991268811, −9.651746363304163927563341244785, −8.350328974521378724886041420000, −7.28485835301333244391798180730, −6.07285150156304793931299153845, −5.48193658757863130595834508340, −3.65301910715398560585064299016, −2.03081524458441067316431322096, 1.88932068965589892498203194377, 3.38691024581919654201952674729, 4.89915561692135464843130782597, 5.73740046134158270187971655643, 6.94204223201256572239402639108, 8.928868186783254686327824693258, 9.877673726346742238787427408557, 10.56672086711209758956905734527, 11.67133201033211112477135999959, 12.74262927039028880413885844167

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