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Properties

Label 2-936-104.69-c1-0-27
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: $\chi_{936} (901, \cdot )$
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

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

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