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Properties

Label 2-912-19.9-c1-0-12
Degree $2$
Conductor $912$
Sign $0.923 + 0.382i$
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  + (0.766 − 0.642i)3-s + (2.20 + 0.802i)5-s + (1.78 − 3.09i)7-s + (0.173 − 0.984i)9-s + (1.35 + 2.35i)11-s + (4.14 + 3.47i)13-s + (2.20 − 0.802i)15-s + (−0.673 − 3.82i)17-s + (−1.01 + 4.23i)19-s + (−0.620 − 3.51i)21-s + (−7.73 + 2.81i)23-s + (0.390 + 0.327i)25-s + (−0.500 − 0.866i)27-s + (0.613 − 3.47i)29-s + (3.26 − 5.65i)31-s + ⋯
L(s)  = 1  + (0.442 − 0.371i)3-s + (0.986 + 0.359i)5-s + (0.675 − 1.16i)7-s + (0.0578 − 0.328i)9-s + (0.409 + 0.709i)11-s + (1.14 + 0.964i)13-s + (0.569 − 0.207i)15-s + (−0.163 − 0.926i)17-s + (−0.231 + 0.972i)19-s + (−0.135 − 0.768i)21-s + (−1.61 + 0.587i)23-s + (0.0781 + 0.0655i)25-s + (−0.0962 − 0.166i)27-s + (0.113 − 0.645i)29-s + (0.586 − 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36933 - 0.471454i\)
\(L(\frac12)\) \(\approx\) \(2.36933 - 0.471454i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (1.01 - 4.23i)T \)
good5 \( 1 + (-2.20 - 0.802i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-1.78 + 3.09i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.35 - 2.35i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.14 - 3.47i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.673 + 3.82i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (7.73 - 2.81i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.613 + 3.47i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-3.26 + 5.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.389T + 37T^{2} \)
41 \( 1 + (1.48 - 1.24i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-4.71 - 1.71i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.518 - 2.94i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (7.80 - 2.84i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.474 + 2.68i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (5.91 - 2.15i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.59 + 14.7i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-8.47 - 3.08i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-7.88 + 6.61i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (9.96 - 8.36i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.08 - 7.07i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.98 - 7.53i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.49 + 8.47i)T + (-91.1 + 33.1i)T^{2} \)
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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

−9.790699208524700471456781143897, −9.503515880773729719836220200401, −8.171258247459167109130063505330, −7.58846219502611489626254530473, −6.54964547510504129503269014009, −6.01409736151798036393196112228, −4.47446856265032892073162926554, −3.80250240178029009434576367535, −2.18871816553686500591652462504, −1.41458045871151046441376280029, 1.51641279786186936437916750209, 2.56524490893463170412062938222, 3.75056119113168975113966378756, 5.00335477013949974122329356669, 5.79392235159287879661408083106, 6.37566858628344552252009580805, 8.065195286174112002610604048791, 8.638219019286980155230973093386, 9.030709569222055355754158626260, 10.16212943773044513666104704106

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