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Properties

Label 2-936-104.101-c1-0-27
Degree $2$
Conductor $936$
Sign $-0.00641 - 0.999i$
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 + i)2-s + 2i·4-s + 3.73·5-s + (−3 − 1.73i)7-s + (−2 + 2i)8-s + (3.73 + 3.73i)10-s + (1 + 1.73i)11-s + (2.59 + 2.5i)13-s + (−1.26 − 4.73i)14-s − 4·16-s + (−0.232 + 0.401i)17-s + (−0.633 + 1.09i)19-s + 7.46i·20-s + (−0.732 + 2.73i)22-s + (4.09 + 7.09i)23-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + 1.66·5-s + (−1.13 − 0.654i)7-s + (−0.707 + 0.707i)8-s + (1.18 + 1.18i)10-s + (0.301 + 0.522i)11-s + (0.720 + 0.693i)13-s + (−0.338 − 1.26i)14-s − 16-s + (−0.0562 + 0.0974i)17-s + (−0.145 + 0.251i)19-s + 1.66i·20-s + (−0.156 + 0.582i)22-s + (0.854 + 1.48i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00641 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00641 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.00641 - 0.999i$
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: \((2,\ 936,\ (\ :1/2),\ -0.00641 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92555 + 1.93793i\)
\(L(\frac12)\) \(\approx\) \(1.92555 + 1.93793i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
13 \( 1 + (-2.59 - 2.5i)T \)
good5 \( 1 - 3.73T + 5T^{2} \)
7 \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.232 - 0.401i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.633 - 1.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.09 - 7.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.59 + 1.5i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.73iT - 31T^{2} \)
37 \( 1 + (2.13 + 3.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.96 + 4.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.19 + 1.26i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.73iT - 47T^{2} \)
53 \( 1 + 3.92iT - 53T^{2} \)
59 \( 1 + (-0.267 + 0.464i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.63 + 6.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.02 + 4.63i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 1.46T + 83T^{2} \)
89 \( 1 + (6.46 - 3.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.19 + 3i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.08412023539536633530782241543, −9.360233796138112197905075528220, −8.772481965723135143715989135779, −7.28157733645469042209151360310, −6.70173035781153376575508480834, −6.03821756743108257369137222180, −5.25660295682820022210260917027, −4.05433764311912467631680730520, −3.10345940500873424179208467540, −1.76012333728908991450330955031, 1.11551106814068224236087518847, 2.59276568702195517920161391102, 3.03812214052929983463667137519, 4.53837728811293886892487644309, 5.70156955878799040095552884800, 6.08952262550183486745177740393, 6.72431351913033562291945935019, 8.667509117010840077712667965580, 9.231200847925196360671801306046, 9.975482804880099460005594972141

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