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Properties

Label 2-156-13.4-c1-0-2
Degree $2$
Conductor $156$
Sign $0.822 + 0.569i$
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  + (−0.5 + 0.866i)3-s − 4.14i·5-s + (2.08 − 1.20i)7-s + (−0.499 − 0.866i)9-s + (3 + 1.73i)11-s + (1.5 + 3.27i)13-s + (3.58 + 2.07i)15-s + (−2.58 − 4.48i)17-s + (−3 + 1.73i)19-s + 2.41i·21-s + (1 − 1.73i)23-s − 12.1·25-s + 0.999·27-s + (−1.58 + 2.75i)29-s − 1.05i·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s − 1.85i·5-s + (0.789 − 0.455i)7-s + (−0.166 − 0.288i)9-s + (0.904 + 0.522i)11-s + (0.416 + 0.909i)13-s + (0.926 + 0.535i)15-s + (−0.628 − 1.08i)17-s + (−0.688 + 0.397i)19-s + 0.526i·21-s + (0.208 − 0.361i)23-s − 2.43·25-s + 0.192·27-s + (−0.295 + 0.511i)29-s − 0.188i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.822 + 0.569i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.822 + 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05690 - 0.330012i\)
\(L(\frac12)\) \(\approx\) \(1.05690 - 0.330012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-1.5 - 3.27i)T \)
good5 \( 1 + 4.14iT - 5T^{2} \)
7 \( 1 + (-2.08 + 1.20i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.58 + 4.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.58 - 2.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.05iT - 31T^{2} \)
37 \( 1 + (-6.58 - 3.80i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.589 + 0.340i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.08 - 10.5i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 - 1.17T + 53T^{2} \)
59 \( 1 + (10.1 - 5.87i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.08 + 1.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 1.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.8iT - 73T^{2} \)
79 \( 1 - 1.82T + 79T^{2} \)
83 \( 1 - 1.36iT - 83T^{2} \)
89 \( 1 + (-6 - 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-15.2 + 8.81i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.72404937857673455885675239814, −11.83246236524359221173862133024, −11.03862328200314933896035232560, −9.432009154217241701066443891018, −9.013706066831028281843662931595, −7.80214958935753081860106907376, −6.21014048053753204435227001131, −4.56106484069067626908446634377, −4.47374449864779133404461512491, −1.38926026282112266105857102187, 2.23080141833620993159800757180, 3.73982971399052110874871265241, 5.80147878610528880964109295369, 6.56059910586342399099871911475, 7.66014644328454915840600066703, 8.756441277280242303651446807199, 10.43548644321582412687364247619, 11.05940989265694986908398089580, 11.75164542981706926327522416233, 13.11306510884583046859183370332

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