Properties

Label 2-912-19.16-c1-0-8
Degree $2$
Conductor $912$
Sign $0.939 - 0.342i$
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 + (0.233 − 1.32i)5-s + (1.20 + 2.08i)7-s + (0.766 − 0.642i)9-s + (−2.97 + 5.14i)11-s + (3.47 + 1.26i)13-s + (−0.233 − 1.32i)15-s + (−0.124 − 0.104i)17-s + (4.11 − 1.43i)19-s + (1.84 + 1.55i)21-s + (1.14 + 6.47i)23-s + (2.99 + 1.08i)25-s + (0.500 − 0.866i)27-s + (1.83 − 1.53i)29-s + (−1.29 − 2.24i)31-s + ⋯
L(s)  = 1  + (0.542 − 0.197i)3-s + (0.104 − 0.593i)5-s + (0.455 + 0.789i)7-s + (0.255 − 0.214i)9-s + (−0.896 + 1.55i)11-s + (0.962 + 0.350i)13-s + (−0.0604 − 0.342i)15-s + (−0.0301 − 0.0253i)17-s + (0.944 − 0.328i)19-s + (0.403 + 0.338i)21-s + (0.238 + 1.34i)23-s + (0.598 + 0.217i)25-s + (0.0962 − 0.166i)27-s + (0.339 − 0.285i)29-s + (−0.233 − 0.403i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.939 - 0.342i$
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: \((2,\ 912,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01297 + 0.355620i\)
\(L(\frac12)\) \(\approx\) \(2.01297 + 0.355620i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.11 + 1.43i)T \)
good5 \( 1 + (-0.233 + 1.32i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-1.20 - 2.08i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.97 - 5.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.47 - 1.26i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.124 + 0.104i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.14 - 6.47i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.83 + 1.53i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.29 + 2.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.94T + 37T^{2} \)
41 \( 1 + (-0.340 + 0.123i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.98 + 11.2i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-5.26 + 4.42i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.698 - 3.96i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (9.23 + 7.74i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.07 - 6.09i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-4.41 + 3.70i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.11 - 12.0i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-8.61 + 3.13i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (9.10 - 3.31i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-1.01 - 1.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.24 + 1.17i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-9.26 - 7.77i)T + (16.8 + 95.5i)T^{2} \)
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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

−9.935784891374414960967339960707, −9.162253475053094421545358783061, −8.597017067171431032097033512752, −7.62498408691928869102542932745, −6.97529002060005528248988891639, −5.52962978989013141320964880394, −4.99982938116745042693977079073, −3.78700849310709285713004863350, −2.45606402740254242538130868334, −1.50532600032258618109653884492, 1.05264543327172294822139512840, 2.83614885236449872481432629461, 3.42773618394178229042746561023, 4.64452785294446608155357250658, 5.72929270644930526260605490237, 6.65675503277141967173321834678, 7.71412056275197438198529533539, 8.308114624153667684798946149967, 9.062400668668903427412275241645, 10.41681834256459263458487423292

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