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Properties

Label 2-912-76.3-c1-0-13
Degree $2$
Conductor $912$
Sign $0.624 + 0.781i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (0.939 − 0.342i)3-s + (−0.134 + 0.764i)5-s + (0.911 − 0.526i)7-s + (0.766 − 0.642i)9-s + (−3.03 − 1.75i)11-s + (1.81 − 4.99i)13-s + (0.134 + 0.764i)15-s + (−0.231 − 0.194i)17-s + (0.105 − 4.35i)19-s + (0.676 − 0.806i)21-s + (5.32 − 0.938i)23-s + (4.13 + 1.50i)25-s + (0.500 − 0.866i)27-s + (1.86 + 2.22i)29-s + (−2.52 − 4.38i)31-s + ⋯
L(s)  = 1  + (0.542 − 0.197i)3-s + (−0.0602 + 0.341i)5-s + (0.344 − 0.198i)7-s + (0.255 − 0.214i)9-s + (−0.916 − 0.529i)11-s + (0.504 − 1.38i)13-s + (0.0347 + 0.197i)15-s + (−0.0561 − 0.0471i)17-s + (0.0243 − 0.999i)19-s + (0.147 − 0.175i)21-s + (1.11 − 0.195i)23-s + (0.826 + 0.300i)25-s + (0.0962 − 0.166i)27-s + (0.346 + 0.412i)29-s + (−0.454 − 0.786i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.624 + 0.781i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.624 + 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68515 - 0.810211i\)
\(L(\frac12)\) \(\approx\) \(1.68515 - 0.810211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.105 + 4.35i)T \)
good5 \( 1 + (0.134 - 0.764i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.911 + 0.526i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.03 + 1.75i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.81 + 4.99i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.231 + 0.194i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-5.32 + 0.938i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.86 - 2.22i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.52 + 4.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.93iT - 37T^{2} \)
41 \( 1 + (-3.67 - 10.0i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (-9.71 - 1.71i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (6.88 + 8.21i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-9.44 + 1.66i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (4.87 + 4.08i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.68 + 9.55i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.02 - 1.69i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.0431 - 0.244i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (4.90 - 1.78i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (1.40 - 0.511i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (7.61 - 4.39i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.89 - 16.1i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-8.49 + 10.1i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.02535657000275559645524803260, −8.993492230067295573447069820928, −8.224346121579994124670178958792, −7.58310710146813250309515352640, −6.68696499788434317467329881210, −5.57009661403805943836544991341, −4.68222468362438204742992742931, −3.24757595049962050121320571204, −2.68745156640764066868318259414, −0.909149534767946967706898712458, 1.56176645476808198251153639621, 2.69633575274384658794053072500, 4.00360511682351588744998544267, 4.78008807271909064340725815187, 5.77024775280948788525753457811, 6.99799933654947968261951172877, 7.72047331948134203243983069908, 8.777166831645559927118692156384, 9.089712961419589269064942426583, 10.24399288207328284357588688558

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