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Properties

Label 2-936-104.101-c1-0-11
Degree $2$
Conductor $936$
Sign $-0.143 - 0.989i$
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  + (−0.579 + 1.29i)2-s + (−1.32 − 1.49i)4-s − 3.41·5-s + (−1.18 − 0.686i)7-s + (2.69 − 0.848i)8-s + (1.97 − 4.40i)10-s + (−3.02 − 5.23i)11-s + (3.56 + 0.533i)13-s + (1.57 − 1.13i)14-s + (−0.467 + 3.97i)16-s + (−3.22 + 5.59i)17-s + (1.47 − 2.56i)19-s + (4.53 + 5.10i)20-s + (8.51 − 0.867i)22-s + (3.18 + 5.51i)23-s + ⋯
L(s)  = 1  + (−0.409 + 0.912i)2-s + (−0.664 − 0.747i)4-s − 1.52·5-s + (−0.449 − 0.259i)7-s + (0.953 − 0.300i)8-s + (0.625 − 1.39i)10-s + (−0.912 − 1.57i)11-s + (0.988 + 0.147i)13-s + (0.421 − 0.303i)14-s + (−0.116 + 0.993i)16-s + (−0.783 + 1.35i)17-s + (0.339 − 0.587i)19-s + (1.01 + 1.14i)20-s + (1.81 − 0.185i)22-s + (0.663 + 1.14i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.143 - 0.989i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.143 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.376622 + 0.434953i\)
\(L(\frac12)\) \(\approx\) \(0.376622 + 0.434953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.579 - 1.29i)T \)
3 \( 1 \)
13 \( 1 + (-3.56 - 0.533i)T \)
good5 \( 1 + 3.41T + 5T^{2} \)
7 \( 1 + (1.18 + 0.686i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.02 + 5.23i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.22 - 5.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.47 + 2.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.18 - 5.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.54 + 0.894i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.78iT - 31T^{2} \)
37 \( 1 + (-2.52 - 4.37i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.96 + 1.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.92 + 2.26i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.675iT - 47T^{2} \)
53 \( 1 - 12.4iT - 53T^{2} \)
59 \( 1 + (-2.78 + 4.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.167 + 0.0966i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.90 - 8.50i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.7 - 6.79i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.97iT - 73T^{2} \)
79 \( 1 + 1.01T + 79T^{2} \)
83 \( 1 + 4.43T + 83T^{2} \)
89 \( 1 + (-11.3 + 6.57i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.0 - 8.70i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.38946273587818958460492481270, −9.031775143399636120221058662423, −8.424451265325950165020724793697, −7.925962903106677698939690786414, −6.96893149195318303289281144260, −6.17742321994939559966064062049, −5.18868929471883088934761949266, −4.00374182729942076474730156397, −3.30666596371037988005752521441, −0.878601119634067493508943794622, 0.45160684632632557764074906616, 2.33441811742864642516762005208, 3.30618287999627986100339043445, 4.32258054134056871334186067776, 4.97621425982099252149079021626, 6.73998495449813726599661290720, 7.59812458818882528190629259594, 8.152832101918500650290268775117, 9.113016847159097579480860931599, 9.856357610437628277556191947249

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