Properties

Label 2-912-19.7-c1-0-8
Degree $2$
Conductor $912$
Sign $0.910 - 0.412i$
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  + (−0.5 − 0.866i)3-s + (2 + 3.46i)5-s + 3·7-s + (−0.499 + 0.866i)9-s − 2·11-s + (3.5 − 6.06i)13-s + (1.99 − 3.46i)15-s + (4 + 1.73i)19-s + (−1.5 − 2.59i)21-s + (−2 + 3.46i)23-s + (−5.49 + 9.52i)25-s + 0.999·27-s + (−2 + 3.46i)29-s − 31-s + (1 + 1.73i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.894 + 1.54i)5-s + 1.13·7-s + (−0.166 + 0.288i)9-s − 0.603·11-s + (0.970 − 1.68i)13-s + (0.516 − 0.894i)15-s + (0.917 + 0.397i)19-s + (−0.327 − 0.566i)21-s + (−0.417 + 0.722i)23-s + (−1.09 + 1.90i)25-s + 0.192·27-s + (−0.371 + 0.643i)29-s − 0.179·31-s + (0.174 + 0.301i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86668 + 0.403440i\)
\(L(\frac12)\) \(\approx\) \(1.86668 + 0.403440i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-4 - 1.73i)T \)
good5 \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-3.5 + 6.06i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + (2 + 3.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (9 - 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.38833755844761193642002767259, −9.530703018723140011124739836572, −8.039838639431400733268567641045, −7.74133044888942073961991935825, −6.70562649621187785759088760674, −5.69570490498245550248519710087, −5.36519544049168526288556332843, −3.51694651573981621049979091414, −2.59239206423703109140038121591, −1.42393030589094712394548124738, 1.13723291485547177989060323285, 2.16441811249907008458530896566, 4.16044528634375194574928402451, 4.72058951128827525089432405681, 5.50860994201189556907059494307, 6.29563231633722245276164898782, 7.72195119982792751824206238234, 8.626699401600791270101429444939, 9.141236924835822165362449789660, 9.866682639141238037181147536627

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