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Properties

Label 2-912-76.15-c1-0-15
Degree $2$
Conductor $912$
Sign $0.447 + 0.894i$
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  + (−0.173 − 0.984i)3-s + (1.30 − 1.09i)5-s + (1.57 + 0.909i)7-s + (−0.939 + 0.342i)9-s + (5.51 − 3.18i)11-s + (−2.14 − 0.378i)13-s + (−1.30 − 1.09i)15-s + (−4.03 − 1.46i)17-s + (4.17 + 1.23i)19-s + (0.622 − 1.70i)21-s + (0.257 − 0.307i)23-s + (−0.368 + 2.08i)25-s + (0.5 + 0.866i)27-s + (1.76 + 4.85i)29-s + (2.11 − 3.66i)31-s + ⋯
L(s)  = 1  + (−0.100 − 0.568i)3-s + (0.581 − 0.487i)5-s + (0.595 + 0.343i)7-s + (−0.313 + 0.114i)9-s + (1.66 − 0.960i)11-s + (−0.596 − 0.105i)13-s + (−0.335 − 0.281i)15-s + (−0.977 − 0.355i)17-s + (0.958 + 0.283i)19-s + (0.135 − 0.373i)21-s + (0.0537 − 0.0640i)23-s + (−0.0736 + 0.417i)25-s + (0.0962 + 0.166i)27-s + (0.327 + 0.900i)29-s + (0.380 − 0.658i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59232 - 0.983161i\)
\(L(\frac12)\) \(\approx\) \(1.59232 - 0.983161i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-4.17 - 1.23i)T \)
good5 \( 1 + (-1.30 + 1.09i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-1.57 - 0.909i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.51 + 3.18i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.14 + 0.378i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (4.03 + 1.46i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-0.257 + 0.307i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-1.76 - 4.85i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.11 + 3.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.939iT - 37T^{2} \)
41 \( 1 + (-7.52 + 1.32i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (3.71 + 4.42i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (3.21 + 8.83i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.589 + 0.702i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-10.8 - 3.95i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (10.5 + 8.87i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (6.48 - 2.36i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (0.701 - 0.588i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (2.33 + 13.2i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-2.36 - 13.3i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.52 + 0.879i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.20 + 0.388i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (1.26 - 3.46i)T + (-74.3 - 62.3i)T^{2} \)
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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

−9.726361294936672197350177457112, −9.010793606396329474277797907321, −8.453469313484768360117448126470, −7.32104535808705688280228368449, −6.48345994046758589203205858627, −5.62747662213426366980632677989, −4.80924642227956388274236621080, −3.49338329275758829471051187774, −2.09169156935255231896809805620, −1.03839872282110784744642389855, 1.52120858130806617546844833849, 2.77474201874205923367611708385, 4.23510070288770983784023054014, 4.65698214624117022451123018954, 6.02962581773069008252884989500, 6.74357764845664845384742199280, 7.60172873522336679967131987173, 8.798528319313584882154230178944, 9.589446400187788985220871866867, 10.05368620519693464829370914498

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