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Properties

Label 2-936-936.155-c1-0-68
Degree $2$
Conductor $936$
Sign $0.842 - 0.538i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
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  + (−1.22 − 0.707i)2-s + (1.72 − 0.190i)3-s + (0.999 + 1.73i)4-s + (−1.00 + 0.583i)5-s + (−2.24 − 0.984i)6-s + (−0.658 + 1.14i)7-s − 2.82i·8-s + (2.92 − 0.654i)9-s + 1.64·10-s + (2.05 + 2.79i)12-s + (1.80 + 3.12i)13-s + (1.61 − 0.931i)14-s + (−1.62 + 1.19i)15-s + (−2.00 + 3.46i)16-s + 1.16i·17-s + (−4.04 − 1.26i)18-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (0.993 − 0.109i)3-s + (0.499 + 0.866i)4-s + (−0.451 + 0.260i)5-s + (−0.915 − 0.401i)6-s + (−0.248 + 0.431i)7-s − 0.999i·8-s + (0.975 − 0.218i)9-s + 0.521·10-s + (0.592 + 0.805i)12-s + (0.499 + 0.866i)13-s + (0.431 − 0.248i)14-s + (−0.420 + 0.308i)15-s + (−0.500 + 0.866i)16-s + 0.283i·17-s + (−0.954 − 0.298i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.842 - 0.538i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26476 + 0.369877i\)
\(L(\frac12)\) \(\approx\) \(1.26476 + 0.369877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-1.72 + 0.190i)T \)
13 \( 1 + (-1.80 - 3.12i)T \)
good5 \( 1 + (1.00 - 0.583i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.658 - 1.14i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 1.16iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.87 - 3.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.39T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.03 - 5.25i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.79 - 2.18i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.75iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (48.5 + 84.0i)T^{2} \)
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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

−9.906670985013756800543417077429, −9.238284436720928160560506257693, −8.598443500601046752753892597220, −7.83271899756069499698239570830, −7.07058437119085900170574483855, −6.20593128290023400234563047284, −4.35422245813665452664938809296, −3.49447748740689687559676771777, −2.59864309906591456718229649342, −1.45162459807940639333613803378, 0.789660692081744442248621593889, 2.31587159292366629406371772272, 3.53523283986068281858260919315, 4.63011148217459124471915299906, 5.85291736484995007241864631375, 6.91990931087476347909942764455, 7.70694207493627705407125418309, 8.279597550781968060719363435123, 8.983489740582713139566230961965, 9.898091512395522724408259541528

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