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Properties

Label 2-1216-19.7-c1-0-25
Degree $2$
Conductor $1216$
Sign $0.188 + 0.982i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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  + (−0.207 − 0.358i)3-s + (−1.60 − 2.78i)5-s + 3.13·7-s + (1.41 − 2.44i)9-s + 2.21·11-s + (−1.37 + 2.37i)13-s + (−0.666 + 1.15i)15-s + (3.59 + 6.22i)17-s + (4.32 + 0.528i)19-s + (−0.649 − 1.12i)21-s + (4.29 − 7.44i)23-s + (−2.67 + 4.63i)25-s − 2.41·27-s + (3.33 − 5.76i)29-s − 10.7·31-s + ⋯
L(s)  = 1  + (−0.119 − 0.207i)3-s + (−0.719 − 1.24i)5-s + 1.18·7-s + (0.471 − 0.816i)9-s + 0.668·11-s + (−0.380 + 0.659i)13-s + (−0.172 + 0.298i)15-s + (0.871 + 1.50i)17-s + (0.992 + 0.121i)19-s + (−0.141 − 0.245i)21-s + (0.896 − 1.55i)23-s + (−0.535 + 0.927i)25-s − 0.464·27-s + (0.618 − 1.07i)29-s − 1.93·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.188 + 0.982i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.188 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.717021214\)
\(L(\frac12)\) \(\approx\) \(1.717021214\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-4.32 - 0.528i)T \)
good3 \( 1 + (0.207 + 0.358i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.60 + 2.78i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.13T + 7T^{2} \)
11 \( 1 - 2.21T + 11T^{2} \)
13 \( 1 + (1.37 - 2.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.59 - 6.22i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.29 + 7.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.33 + 5.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 - 0.308T + 37T^{2} \)
41 \( 1 + (-2.30 - 3.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.31 + 7.47i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.47 + 4.27i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.00 - 6.93i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.01 - 1.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.780 - 1.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.81 + 6.60i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.84 - 8.38i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.85 + 8.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.98 + 5.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.49T + 83T^{2} \)
89 \( 1 + (-0.762 + 1.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.69 + 4.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.301535457605779772144569866762, −8.722430305193492962195781723539, −7.984898886156148638896328981470, −7.24702803883987820248299912367, −6.18875505113365203619950837030, −5.12932850206731494305635253518, −4.36237253892185539091841541358, −3.69063806778335696100298226930, −1.73263307495946582746159028290, −0.893936413356933852554455562173, 1.40883564204202788736398537727, 2.89733049149527614292527194953, 3.65248162948557282318693581184, 4.99475841098943339078066497974, 5.35593298092952858711846150484, 7.02230209501460233398819286929, 7.42912520227188985825060297078, 7.916365964078893215337344039027, 9.243744700964732755212946916374, 9.966865859317366464926793047934

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