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Properties

Label 2-912-12.11-c1-0-29
Degree $2$
Conductor $912$
Sign $0.117 + 0.993i$
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  + (1.72 − 0.203i)3-s − 2.74i·5-s − 1.91i·7-s + (2.91 − 0.699i)9-s − 0.463·11-s + 0.406·13-s + (−0.557 − 4.71i)15-s + 0.415i·17-s + i·19-s + (−0.389 − 3.29i)21-s − 2.91·23-s − 2.51·25-s + (4.87 − 1.79i)27-s − 7.11i·29-s + 2.40i·31-s + ⋯
L(s)  = 1  + (0.993 − 0.117i)3-s − 1.22i·5-s − 0.724i·7-s + (0.972 − 0.233i)9-s − 0.139·11-s + 0.112·13-s + (−0.143 − 1.21i)15-s + 0.100i·17-s + 0.229i·19-s + (−0.0850 − 0.719i)21-s − 0.608·23-s − 0.502·25-s + (0.938 − 0.345i)27-s − 1.32i·29-s + 0.432i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.117 + 0.993i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.117 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61901 - 1.43887i\)
\(L(\frac12)\) \(\approx\) \(1.61901 - 1.43887i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.203i)T \)
19 \( 1 - iT \)
good5 \( 1 + 2.74iT - 5T^{2} \)
7 \( 1 + 1.91iT - 7T^{2} \)
11 \( 1 + 0.463T + 11T^{2} \)
13 \( 1 - 0.406T + 13T^{2} \)
17 \( 1 - 0.415iT - 17T^{2} \)
23 \( 1 + 2.91T + 23T^{2} \)
29 \( 1 + 7.11iT - 29T^{2} \)
31 \( 1 - 2.40iT - 31T^{2} \)
37 \( 1 + 6.61T + 37T^{2} \)
41 \( 1 + 5.43iT - 41T^{2} \)
43 \( 1 - 5.91iT - 43T^{2} \)
47 \( 1 - 3.61T + 47T^{2} \)
53 \( 1 + 9.15iT - 53T^{2} \)
59 \( 1 + 7.99T + 59T^{2} \)
61 \( 1 - 8.15T + 61T^{2} \)
67 \( 1 - 14.8iT - 67T^{2} \)
71 \( 1 - 9.75T + 71T^{2} \)
73 \( 1 + 1.13T + 73T^{2} \)
79 \( 1 - 13.8iT - 79T^{2} \)
83 \( 1 - 6.54T + 83T^{2} \)
89 \( 1 - 6.83iT - 89T^{2} \)
97 \( 1 - 11.2T + 97T^{2} \)
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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

−9.801762069273734345634114923460, −8.929856192092183462676017507097, −8.291117541603577782092705597455, −7.61199425799975665159499174470, −6.63299351429930284846873954980, −5.36669443442575253299279799958, −4.34511090828822192137142182082, −3.65456923535477343530658812361, −2.18047925333832605415611028399, −0.958306744459445286968802669953, 1.97620129794281148953160450468, 2.88965145344656007627973464304, 3.64228224744364207638935585751, 4.92824509186731988193704553722, 6.13299755635299302229829938943, 7.02517279631721879892760378200, 7.73199310238707304401304454283, 8.709152877301956849518245679576, 9.338151135125757422184677343520, 10.34786718125987243765393172311

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