Properties

Label 2-912-57.29-c1-0-18
Degree $2$
Conductor $912$
Sign $0.943 + 0.331i$
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  + (−0.517 − 1.65i)3-s + (−0.258 + 0.710i)5-s + (−0.777 + 1.34i)7-s + (−2.46 + 1.71i)9-s + (−0.832 + 0.480i)11-s + (0.416 − 0.496i)13-s + (1.30 + 0.0594i)15-s + (6.73 − 1.18i)17-s + (4.14 + 1.35i)19-s + (2.62 + 0.587i)21-s + (−0.400 − 1.10i)23-s + (3.39 + 2.84i)25-s + (4.10 + 3.18i)27-s + (1.39 − 7.92i)29-s + (2.63 + 1.52i)31-s + ⋯
L(s)  = 1  + (−0.298 − 0.954i)3-s + (−0.115 + 0.317i)5-s + (−0.294 + 0.509i)7-s + (−0.821 + 0.570i)9-s + (−0.250 + 0.144i)11-s + (0.115 − 0.137i)13-s + (0.337 + 0.0153i)15-s + (1.63 − 0.287i)17-s + (0.950 + 0.310i)19-s + (0.573 + 0.128i)21-s + (−0.0835 − 0.229i)23-s + (0.678 + 0.569i)25-s + (0.790 + 0.613i)27-s + (0.259 − 1.47i)29-s + (0.474 + 0.273i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.943 + 0.331i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.943 + 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32018 - 0.225153i\)
\(L(\frac12)\) \(\approx\) \(1.32018 - 0.225153i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.517 + 1.65i)T \)
19 \( 1 + (-4.14 - 1.35i)T \)
good5 \( 1 + (0.258 - 0.710i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.777 - 1.34i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.832 - 0.480i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.416 + 0.496i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-6.73 + 1.18i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.400 + 1.10i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.39 + 7.92i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.63 - 1.52i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.12iT - 37T^{2} \)
41 \( 1 + (4.09 - 3.43i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-7.34 - 2.67i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-3.11 - 0.548i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (13.6 - 4.96i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (2.02 + 11.4i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-10.1 + 3.70i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-9.19 - 1.62i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (0.0322 + 0.0117i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (3.04 - 2.55i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-0.893 - 1.06i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-10.4 - 6.05i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.68 - 3.92i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-9.54 + 1.68i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.03558095094437279438624624507, −9.253173207816609910018959126456, −7.995066231742386893331656247724, −7.66062004112348064216031127299, −6.58771667783585818249801581347, −5.81185213332554830543762744528, −5.03971836095829953459849780251, −3.38339122681935175570616137462, −2.50818123410924002530101783744, −1.01359584631504499870467077252, 0.922094478915827211841708521402, 3.03978612073518199256886213844, 3.78029957000221125720189274883, 4.91090096967674792948555296030, 5.55270056769120157069199032543, 6.63430457355892248653077891766, 7.67960770063572265266689858662, 8.609301329517416446896861423900, 9.426475936777769209904481096556, 10.23360457234211096732868830321

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