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Properties

Label 2-1216-19.11-c1-0-13
Degree $2$
Conductor $1216$
Sign $0.937 - 0.347i$
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  + (1.20 − 2.09i)3-s + (−1.06 + 1.84i)5-s − 1.59·7-s + (−1.41 − 2.44i)9-s + 1.12·11-s + (3.27 + 5.67i)13-s + (2.56 + 4.45i)15-s + (−2.14 + 3.72i)17-s + (2.69 + 3.42i)19-s + (−1.92 + 3.33i)21-s + (−2.85 − 4.94i)23-s + (0.233 + 0.404i)25-s + 0.414·27-s + (0.882 + 1.52i)29-s + 5.25·31-s + ⋯
L(s)  = 1  + (0.696 − 1.20i)3-s + (−0.476 + 0.824i)5-s − 0.603·7-s + (−0.471 − 0.816i)9-s + 0.340·11-s + (0.908 + 1.57i)13-s + (0.663 + 1.14i)15-s + (−0.520 + 0.902i)17-s + (0.617 + 0.786i)19-s + (−0.420 + 0.728i)21-s + (−0.595 − 1.03i)23-s + (0.0467 + 0.0809i)25-s + 0.0797·27-s + (0.163 + 0.283i)29-s + 0.943·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.937 - 0.347i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.937 - 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.755827861\)
\(L(\frac12)\) \(\approx\) \(1.755827861\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.69 - 3.42i)T \)
good3 \( 1 + (-1.20 + 2.09i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.06 - 1.84i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.59T + 7T^{2} \)
11 \( 1 - 1.12T + 11T^{2} \)
13 \( 1 + (-3.27 - 5.67i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.14 - 3.72i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.85 + 4.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.882 - 1.52i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.25T + 31T^{2} \)
37 \( 1 - 1.23T + 37T^{2} \)
41 \( 1 + (-4.04 + 7.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.35 - 4.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.973 - 1.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.56 - 7.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.33 + 2.31i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.89 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.87 - 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.71 + 4.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.967 + 1.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.49 - 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.06T + 83T^{2} \)
89 \( 1 + (-0.680 - 1.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.00 - 15.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.641159621307258439804831715452, −8.721955682077209871415254579185, −8.143331444735239822970156412410, −7.20031770721882813243207844850, −6.57872659928469507595561036935, −6.14189035115509850990292189610, −4.26715230642203893102455239693, −3.48413169980041928593711717316, −2.43315907263584100692859734101, −1.40034248160446331180875069100, 0.75663905149875832281039025593, 2.85084389742012291569812459291, 3.51257378547856897771558067468, 4.42629633650535298281055107512, 5.17433573668628605987393712898, 6.21254439782630563136739484338, 7.46408288927450835902735522093, 8.379583606983479787986932989580, 8.861123868368239735503968584460, 9.708955931585857847260835652739

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