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Properties

Label 2-156-156.107-c1-0-0
Degree $2$
Conductor $156$
Sign $-0.907 - 0.419i$
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  + (0.707 + 1.22i)2-s + (−1.72 + 0.178i)3-s + (−0.999 + 1.73i)4-s + 2.23i·5-s + (−1.43 − 1.98i)6-s + (−2.73 − 1.58i)7-s − 2.82·8-s + (2.93 − 0.614i)9-s + (−2.73 + 1.58i)10-s + (2.12 + 3.67i)11-s + (1.41 − 3.16i)12-s + (−3.5 + 0.866i)13-s − 4.47i·14-s + (−0.398 − 3.85i)15-s + (−2.00 − 3.46i)16-s + (1.93 + 1.11i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (−0.994 + 0.102i)3-s + (−0.499 + 0.866i)4-s + 0.999i·5-s + (−0.586 − 0.809i)6-s + (−1.03 − 0.597i)7-s − 0.999·8-s + (0.978 − 0.204i)9-s + (−0.866 + 0.500i)10-s + (0.639 + 1.10i)11-s + (0.408 − 0.912i)12-s + (−0.970 + 0.240i)13-s − 1.19i·14-s + (−0.102 − 0.994i)15-s + (−0.500 − 0.866i)16-s + (0.469 + 0.271i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.907 - 0.419i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ -0.907 - 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180213 + 0.818649i\)
\(L(\frac12)\) \(\approx\) \(0.180213 + 0.818649i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 + (1.72 - 0.178i)T \)
13 \( 1 + (3.5 - 0.866i)T \)
good5 \( 1 - 2.23iT - 5T^{2} \)
7 \( 1 + (2.73 + 1.58i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.12 - 3.67i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.93 - 1.11i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.47 - 3.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.41 - 2.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.80 + 3.35i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.16iT - 31T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.68 - 5.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.73 - 1.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 2.23iT - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.47 - 3.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.41 - 2.44i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 + 12.6iT - 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + (3.87 - 2.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8 + 13.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.41689762235597094771924293875, −12.33530472273834310787695503905, −11.72012451165303557408384715027, −10.09991823742183288136919979510, −9.670590888379472286760873913929, −7.44339454142121572632951242881, −6.94143386963254968350406030603, −6.04508300965237779054165359083, −4.65156705004792126543715859350, −3.39308774681080973759350447726, 0.821068812612578224845769304164, 3.14753407433093125746151939490, 4.85659907882108061838096849537, 5.58374306567673702825515917550, 6.76323390967179705387764369362, 8.822026341216631218685880451010, 9.617324630957235483079609688916, 10.66353094055424409888938575821, 12.00167287587216319765346794641, 12.17859895067792940895915774764

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