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Properties

Label 2-156-156.35-c1-0-16
Degree $2$
Conductor $156$
Sign $0.923 + 0.383i$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 + 0.187i)2-s + (−1.64 − 0.539i)3-s + (1.92 + 0.526i)4-s − 1.98i·5-s + (−2.20 − 1.06i)6-s + (2.42 − 1.40i)7-s + (2.60 + 1.10i)8-s + (2.41 + 1.77i)9-s + (0.372 − 2.77i)10-s + (−2.01 + 3.49i)11-s + (−2.89 − 1.90i)12-s + (0.235 − 3.59i)13-s + (3.66 − 1.50i)14-s + (−1.06 + 3.26i)15-s + (3.44 + 2.03i)16-s + (−6.77 + 3.91i)17-s + ⋯
L(s)  = 1  + (0.991 + 0.132i)2-s + (−0.950 − 0.311i)3-s + (0.964 + 0.263i)4-s − 0.885i·5-s + (−0.900 − 0.434i)6-s + (0.916 − 0.529i)7-s + (0.921 + 0.389i)8-s + (0.805 + 0.591i)9-s + (0.117 − 0.878i)10-s + (−0.607 + 1.05i)11-s + (−0.834 − 0.550i)12-s + (0.0652 − 0.997i)13-s + (0.978 − 0.402i)14-s + (−0.275 + 0.841i)15-s + (0.861 + 0.508i)16-s + (−1.64 + 0.948i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.923 + 0.383i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55903 - 0.310559i\)
\(L(\frac12)\) \(\approx\) \(1.55903 - 0.310559i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.187i)T \)
3 \( 1 + (1.64 + 0.539i)T \)
13 \( 1 + (-0.235 + 3.59i)T \)
good5 \( 1 + 1.98iT - 5T^{2} \)
7 \( 1 + (-2.42 + 1.40i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.01 - 3.49i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (6.77 - 3.91i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.36 - 1.94i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.939 + 1.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.74 + 1.00i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.09iT - 31T^{2} \)
37 \( 1 + (2.61 - 4.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.61 - 2.08i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.06 + 1.19i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.72T + 47T^{2} \)
53 \( 1 + 1.24iT - 53T^{2} \)
59 \( 1 + (0.420 + 0.729i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.28 + 2.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.50 + 3.75i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.46 + 12.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.63T + 73T^{2} \)
79 \( 1 + 7.13iT - 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 + (2.80 + 1.61i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.30 + 10.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.80236591040284418236807570707, −12.25655232611504604014717863105, −10.92543814744327880968881341144, −10.48588159706023395295507400805, −8.351035762564552635419898140149, −7.40214933359969303066614574662, −6.20035249646673452200024503537, −4.92519551080202558620419410811, −4.44089991555344446083653941238, −1.81161065704268665113910200495, 2.43119498668162731261968920431, 4.20121808469175137726327348737, 5.26219912382360758983068934303, 6.33912377517680291461856005145, 7.23268331817566379192075003934, 9.018044962284188906185727494235, 10.68841794308105218105222962559, 11.17588248165114776467294242287, 11.67852754353718570423739722151, 13.02304380088179986637477516563

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