Properties

Label 2-912-57.41-c1-0-9
Degree $2$
Conductor $912$
Sign $-0.689 - 0.723i$
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  + (0.845 + 1.51i)3-s + (−2.22 − 0.392i)5-s + (1.16 + 2.02i)7-s + (−1.57 + 2.55i)9-s + (2.52 + 1.45i)11-s + (−0.451 + 1.24i)13-s + (−1.28 − 3.70i)15-s + (3.72 − 4.43i)17-s + (1.79 + 3.97i)19-s + (−2.07 + 3.47i)21-s + (−8.06 + 1.42i)23-s + (0.112 + 0.0411i)25-s + (−5.19 − 0.213i)27-s + (−1.64 + 1.38i)29-s + (−5.27 + 3.04i)31-s + ⋯
L(s)  = 1  + (0.488 + 0.872i)3-s + (−0.996 − 0.175i)5-s + (0.441 + 0.764i)7-s + (−0.523 + 0.852i)9-s + (0.760 + 0.438i)11-s + (−0.125 + 0.344i)13-s + (−0.333 − 0.955i)15-s + (0.902 − 1.07i)17-s + (0.412 + 0.910i)19-s + (−0.451 + 0.758i)21-s + (−1.68 + 0.296i)23-s + (0.0225 + 0.00822i)25-s + (−0.999 − 0.0410i)27-s + (−0.305 + 0.256i)29-s + (−0.948 + 0.547i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.689 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.689 - 0.723i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.689 - 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.521600 + 1.21741i\)
\(L(\frac12)\) \(\approx\) \(0.521600 + 1.21741i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.845 - 1.51i)T \)
19 \( 1 + (-1.79 - 3.97i)T \)
good5 \( 1 + (2.22 + 0.392i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.16 - 2.02i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.52 - 1.45i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.451 - 1.24i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.72 + 4.43i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (8.06 - 1.42i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.64 - 1.38i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (5.27 - 3.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.98iT - 37T^{2} \)
41 \( 1 + (8.52 - 3.10i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.0666 + 0.377i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-6.57 - 7.83i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.494 - 2.80i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (2.53 + 2.12i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.01 - 5.77i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-10.4 - 12.4i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.29 + 13.0i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-5.84 + 2.12i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (1.77 + 4.87i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.62 - 0.938i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.82 - 2.11i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-6.13 + 7.30i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.17926903841955637793438353308, −9.555790967540494814812569662910, −8.756688803774619034873317535744, −7.977601946264935425354407189485, −7.34433796019687375755854917806, −5.84704805903547467041979806292, −4.97520686918504399462271774586, −4.05728216131631747446698739809, −3.30974170162815469526583934812, −1.88390795180595466555681191935, 0.59680101632459165713227544202, 1.97833117728868037744716952191, 3.59494262152671644023541700645, 3.91980446366279062625717659102, 5.54773327580180972848075565086, 6.56933724099000271626188555483, 7.46159716886708865087249681613, 7.938537340403672380343326635427, 8.644261980769435770138718063440, 9.732040467581473721150499950264

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