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Properties

Label 2-156-4.3-c2-0-0
Degree $2$
Conductor $156$
Sign $-0.251 + 0.967i$
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  + (−0.253 + 1.98i)2-s + 1.73i·3-s + (−3.87 − 1.00i)4-s − 5.54·5-s + (−3.43 − 0.438i)6-s + 3.35i·7-s + (2.97 − 7.42i)8-s − 2.99·9-s + (1.40 − 10.9i)10-s − 14.0i·11-s + (1.73 − 6.70i)12-s − 3.60·13-s + (−6.65 − 0.849i)14-s − 9.60i·15-s + (13.9 + 7.77i)16-s − 24.9·17-s + ⋯
L(s)  = 1  + (−0.126 + 0.991i)2-s + 0.577i·3-s + (−0.967 − 0.251i)4-s − 1.10·5-s + (−0.572 − 0.0730i)6-s + 0.479i·7-s + (0.371 − 0.928i)8-s − 0.333·9-s + (0.140 − 1.09i)10-s − 1.27i·11-s + (0.144 − 0.558i)12-s − 0.277·13-s + (−0.475 − 0.0606i)14-s − 0.640i·15-s + (0.873 + 0.486i)16-s − 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.251 + 0.967i$
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: \((2,\ 156,\ (\ :1),\ -0.251 + 0.967i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0770662 - 0.0996110i\)
\(L(\frac12)\) \(\approx\) \(0.0770662 - 0.0996110i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.253 - 1.98i)T \)
3 \( 1 - 1.73iT \)
13 \( 1 + 3.60T \)
good5 \( 1 + 5.54T + 25T^{2} \)
7 \( 1 - 3.35iT - 49T^{2} \)
11 \( 1 + 14.0iT - 121T^{2} \)
17 \( 1 + 24.9T + 289T^{2} \)
19 \( 1 - 11.7iT - 361T^{2} \)
23 \( 1 + 13.4iT - 529T^{2} \)
29 \( 1 + 10.6T + 841T^{2} \)
31 \( 1 - 43.3iT - 961T^{2} \)
37 \( 1 + 42.7T + 1.36e3T^{2} \)
41 \( 1 + 42.3T + 1.68e3T^{2} \)
43 \( 1 + 32.6iT - 1.84e3T^{2} \)
47 \( 1 - 62.4iT - 2.20e3T^{2} \)
53 \( 1 + 47.2T + 2.80e3T^{2} \)
59 \( 1 - 62.1iT - 3.48e3T^{2} \)
61 \( 1 - 93.4T + 3.72e3T^{2} \)
67 \( 1 + 19.2iT - 4.48e3T^{2} \)
71 \( 1 - 22.3iT - 5.04e3T^{2} \)
73 \( 1 + 125.T + 5.32e3T^{2} \)
79 \( 1 - 67.4iT - 6.24e3T^{2} \)
83 \( 1 + 99.8iT - 6.88e3T^{2} \)
89 \( 1 - 8.71T + 7.92e3T^{2} \)
97 \( 1 - 21.1T + 9.40e3T^{2} \)
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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

−13.70539854971424706673438408890, −12.44228569198645908582024340867, −11.33321466804705869081089886429, −10.32083334123764675174477977951, −8.800954812146335148083444062460, −8.478784647704295708699601082753, −7.13122269596785256215987482507, −5.89457159820049168912294312796, −4.66284207921310091150815381750, −3.50447405511009945129431459491, 0.081362665368267450819684090455, 2.10339824860220224979035021448, 3.81423640929657773046884940345, 4.83173405602426088645698694258, 6.98420719741670086239524298009, 7.80618721992550598920291858321, 8.956912687884601032058729200311, 10.10585766636032406752625628040, 11.30330413313561333538880257851, 11.82824757183919236220445126124

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