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Properties

Label 2-912-57.8-c1-0-8
Degree $2$
Conductor $912$
Sign $-0.678 - 0.734i$
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  + (−1.73 + 0.0649i)3-s + (3.15 + 1.82i)5-s − 1.32·7-s + (2.99 − 0.224i)9-s + 5.22i·11-s + (−5.77 + 3.33i)13-s + (−5.58 − 2.95i)15-s + (−4.40 − 2.54i)17-s + (3.57 − 2.49i)19-s + (2.29 − 0.0862i)21-s + (2.14 − 1.23i)23-s + (4.14 + 7.18i)25-s + (−5.16 + 0.583i)27-s + (−0.559 − 0.969i)29-s + 0.304i·31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0375i)3-s + (1.41 + 0.815i)5-s − 0.501·7-s + (0.997 − 0.0749i)9-s + 1.57i·11-s + (−1.60 + 0.925i)13-s + (−1.44 − 0.761i)15-s + (−1.06 − 0.616i)17-s + (0.819 − 0.573i)19-s + (0.501 − 0.0188i)21-s + (0.447 − 0.258i)23-s + (0.829 + 1.43i)25-s + (−0.993 + 0.112i)27-s + (−0.103 − 0.179i)29-s + 0.0546i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.678 - 0.734i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.357532 + 0.817432i\)
\(L(\frac12)\) \(\approx\) \(0.357532 + 0.817432i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0649i)T \)
19 \( 1 + (-3.57 + 2.49i)T \)
good5 \( 1 + (-3.15 - 1.82i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 1.32T + 7T^{2} \)
11 \( 1 - 5.22iT - 11T^{2} \)
13 \( 1 + (5.77 - 3.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.40 + 2.54i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.14 + 1.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.559 + 0.969i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.304iT - 31T^{2} \)
37 \( 1 + 4.66iT - 37T^{2} \)
41 \( 1 + (2.16 - 3.74i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.93 - 8.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.04 - 4.06i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.391 + 0.677i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.58 + 4.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.21 - 12.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.10 - 1.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.10 + 3.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.88 - 8.45i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.31 + 2.48i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.31iT - 83T^{2} \)
89 \( 1 + (0.227 + 0.393i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.45 - 5.45i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.13908562010536307730872639410, −9.701891609532178767052443846030, −9.348832476413432131881488974465, −7.23579251992143305236718549901, −6.96226659978601486121807133541, −6.27204692366203095719965742739, −5.06331490587540006119678584348, −4.58700124195986056511206182630, −2.70041461731110632902562148540, −1.85997976618215693615521599013, 0.45272519688355431143213806055, 1.84634558010274132588737603068, 3.29442377552508395077104826587, 4.90049897699526813560169716651, 5.44214365133372321320287340272, 6.10003765088651987116201373679, 6.94855665970077513576765086381, 8.203653686283906856476510789993, 9.131843391353166212405883850678, 9.968048253166420875304580267412

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