Properties

Label 2-1216-19.18-c2-0-49
Degree $2$
Conductor $1216$
Sign $0.315 + 0.948i$
Analytic cond. $33.1336$
Root an. cond. $5.75617$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.60i·3-s − 4·5-s + 5·7-s − 3.99·9-s − 10·11-s + 3.60i·13-s − 14.4i·15-s + 15·17-s + (−6 − 18.0i)19-s + 18.0i·21-s − 35·23-s − 9·25-s + 18.0i·27-s + 18.0i·29-s − 36.0i·31-s + ⋯
L(s)  = 1  + 1.20i·3-s − 0.800·5-s + 0.714·7-s − 0.444·9-s − 0.909·11-s + 0.277i·13-s − 0.961i·15-s + 0.882·17-s + (−0.315 − 0.948i)19-s + 0.858i·21-s − 1.52·23-s − 0.359·25-s + 0.667i·27-s + 0.621i·29-s − 1.16i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.315 + 0.948i$
Analytic conductor: \(33.1336\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1),\ 0.315 + 0.948i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5839130913\)
\(L(\frac12)\) \(\approx\) \(0.5839130913\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (6 + 18.0i)T \)
good3 \( 1 - 3.60iT - 9T^{2} \)
5 \( 1 + 4T + 25T^{2} \)
7 \( 1 - 5T + 49T^{2} \)
11 \( 1 + 10T + 121T^{2} \)
13 \( 1 - 3.60iT - 169T^{2} \)
17 \( 1 - 15T + 289T^{2} \)
23 \( 1 + 35T + 529T^{2} \)
29 \( 1 - 18.0iT - 841T^{2} \)
31 \( 1 + 36.0iT - 961T^{2} \)
37 \( 1 + 21.6iT - 1.36e3T^{2} \)
41 \( 1 + 36.0iT - 1.68e3T^{2} \)
43 \( 1 + 20T + 1.84e3T^{2} \)
47 \( 1 + 10T + 2.20e3T^{2} \)
53 \( 1 + 75.7iT - 2.80e3T^{2} \)
59 \( 1 + 18.0iT - 3.48e3T^{2} \)
61 \( 1 - 40T + 3.72e3T^{2} \)
67 \( 1 + 39.6iT - 4.48e3T^{2} \)
71 \( 1 - 108. iT - 5.04e3T^{2} \)
73 \( 1 - 105T + 5.32e3T^{2} \)
79 \( 1 + 36.0iT - 6.24e3T^{2} \)
83 \( 1 + 40T + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 122. iT - 9.40e3T^{2} \)
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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

−9.589567326250622356930278152546, −8.471060404125945718378618311284, −7.935097899905143739865265091968, −7.08250472108462689397100210186, −5.69627482715020209623018994046, −4.94434417578348280169650173334, −4.17253266087763365785599155327, −3.46198570509865113918864939610, −2.08797418286486006571866143206, −0.18093373546281496422424141430, 1.20854645194228079265580948691, 2.20983681933586124752378825093, 3.47817550771349350694678232705, 4.55960095006345851383012882988, 5.62871307630947176080034633845, 6.45442594173889976787522414484, 7.58252707996188553207210961000, 7.953874850726800222253321761387, 8.313943586375826253115996016706, 9.855705431160310149988817256651

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