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Properties

Label 2-912-912.221-c1-0-17
Degree $2$
Conductor $912$
Sign $0.994 + 0.105i$
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  + (−0.788 − 1.17i)2-s + (−0.920 − 1.46i)3-s + (−0.755 + 1.85i)4-s + (−0.395 + 1.47i)5-s + (−0.996 + 2.23i)6-s − 2.65i·7-s + (2.76 − 0.574i)8-s + (−1.30 + 2.70i)9-s + (2.04 − 0.699i)10-s + (−1.98 − 1.98i)11-s + (3.41 − 0.596i)12-s + (0.498 + 1.85i)13-s + (−3.11 + 2.09i)14-s + (2.52 − 0.777i)15-s + (−2.85 − 2.79i)16-s + (−4.07 + 2.35i)17-s + ⋯
L(s)  = 1  + (−0.557 − 0.829i)2-s + (−0.531 − 0.847i)3-s + (−0.377 + 0.925i)4-s + (−0.176 + 0.659i)5-s + (−0.406 + 0.913i)6-s − 1.00i·7-s + (0.979 − 0.203i)8-s + (−0.435 + 0.900i)9-s + (0.645 − 0.221i)10-s + (−0.598 − 0.598i)11-s + (0.985 − 0.172i)12-s + (0.138 + 0.515i)13-s + (−0.831 + 0.559i)14-s + (0.652 − 0.200i)15-s + (−0.714 − 0.699i)16-s + (−0.987 + 0.570i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.994 + 0.105i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.994 + 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613892 - 0.0326272i\)
\(L(\frac12)\) \(\approx\) \(0.613892 - 0.0326272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.788 + 1.17i)T \)
3 \( 1 + (0.920 + 1.46i)T \)
19 \( 1 + (-1.44 - 4.11i)T \)
good5 \( 1 + (0.395 - 1.47i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + 2.65iT - 7T^{2} \)
11 \( 1 + (1.98 + 1.98i)T + 11iT^{2} \)
13 \( 1 + (-0.498 - 1.85i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.07 - 2.35i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.221 + 0.383i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.98 - 1.06i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 - 1.03iT - 31T^{2} \)
37 \( 1 + (-2.98 - 2.98i)T + 37iT^{2} \)
41 \( 1 + (-3.48 + 2.01i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.46 - 1.73i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.11 - 1.22i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.42 - 9.03i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.18 + 1.65i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.42 - 0.649i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.35 - 5.07i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.01 + 5.20i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-12.4 + 7.17i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.36 - 5.40i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.23 - 2.23i)T - 83iT^{2} \)
89 \( 1 + (-0.103 - 0.0594i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.17 - 1.83i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.62718401578388195514978832408, −9.323149791321097778647179407375, −8.287867113493996239994299136075, −7.56186226072891942432530785252, −6.94749191609472587626537303937, −5.93267788601070893575930871082, −4.52307801242111796763580587563, −3.49200228325284715026796758019, −2.32728898524635559127133998668, −1.06518483377951113885091966912, 0.45858807606721427281541045087, 2.49772402264535163776906381590, 4.28280922738935118674736948440, 5.07381519676073533651628573250, 5.59571355054446869390493960346, 6.61295402468039563604255514402, 7.64486511522409388430563991300, 8.690557516348951310465075654521, 9.162536766087266266717123345597, 9.819235030140938864125450983151

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