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Properties

Label 2-912-19.16-c1-0-5
Degree $2$
Conductor $912$
Sign $0.0449 - 0.998i$
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 + (−0.554 + 3.14i)5-s + (2.51 + 4.36i)7-s + (0.766 − 0.642i)9-s + (1.30 − 2.25i)11-s + (0.271 + 0.0989i)13-s + (0.554 + 3.14i)15-s + (−5.14 − 4.31i)17-s + (−0.261 + 4.35i)19-s + (3.85 + 3.23i)21-s + (0.502 + 2.85i)23-s + (−4.87 − 1.77i)25-s + (0.500 − 0.866i)27-s + (−0.423 + 0.354i)29-s + (1.38 + 2.39i)31-s + ⋯
L(s)  = 1  + (0.542 − 0.197i)3-s + (−0.247 + 1.40i)5-s + (0.952 + 1.64i)7-s + (0.255 − 0.214i)9-s + (0.392 − 0.679i)11-s + (0.0754 + 0.0274i)13-s + (0.143 + 0.811i)15-s + (−1.24 − 1.04i)17-s + (−0.0600 + 0.998i)19-s + (0.842 + 0.706i)21-s + (0.104 + 0.594i)23-s + (−0.975 − 0.355i)25-s + (0.0962 − 0.166i)27-s + (−0.0785 + 0.0659i)29-s + (0.248 + 0.430i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0449 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0449 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.0449 - 0.998i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.0449 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37560 + 1.31509i\)
\(L(\frac12)\) \(\approx\) \(1.37560 + 1.31509i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.261 - 4.35i)T \)
good5 \( 1 + (0.554 - 3.14i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-2.51 - 4.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.30 + 2.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.271 - 0.0989i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (5.14 + 4.31i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.502 - 2.85i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.423 - 0.354i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.38 - 2.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 + (-7.80 + 2.84i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.586 - 3.32i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-8.22 + 6.90i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.0123 - 0.0699i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-3.86 - 3.24i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.550 - 3.12i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-9.18 + 7.70i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.654 + 3.71i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-9.52 + 3.46i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (4.70 - 1.71i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (5.45 + 9.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.20 - 1.53i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-7.68 - 6.44i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.41400407118759992930692956256, −9.164261375605559929857390343106, −8.712212321189899046449816961604, −7.80113399290133176181350353972, −6.93971684530290712953734887251, −6.07854746283251372711220713929, −5.13235848878687679617887110982, −3.71684402496474787217986141000, −2.75290901083391322819180356654, −1.96605304012156984473080703686, 0.884277457200542746216670392945, 2.01100036147581823553821394205, 4.00668000063885255115044588429, 4.33610936618950631829717570849, 5.08261945444944377394398809425, 6.71045568294742837132338320266, 7.47985812247191548291099540899, 8.391163756583891757996212203462, 8.827633036635819392466621984393, 9.827336089647770100310301617261

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