Properties

Label 2-1216-19.11-c1-0-37
Degree $2$
Conductor $1216$
Sign $-0.936 - 0.350i$
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  + (1.62 − 2.81i)3-s + (0.235 − 0.407i)5-s − 3.30·7-s + (−3.77 − 6.54i)9-s − 1.47·11-s + (−1.41 − 2.45i)13-s + (−0.764 − 1.32i)15-s + (−3.35 + 5.81i)17-s + (0.206 − 4.35i)19-s + (−5.37 + 9.30i)21-s + (−0.235 − 0.407i)23-s + (2.38 + 4.13i)25-s − 14.8·27-s + (2.48 + 4.30i)29-s + 6.74·31-s + ⋯
L(s)  = 1  + (0.937 − 1.62i)3-s + (0.105 − 0.182i)5-s − 1.25·7-s + (−1.25 − 2.18i)9-s − 0.443·11-s + (−0.393 − 0.681i)13-s + (−0.197 − 0.341i)15-s + (−0.814 + 1.41i)17-s + (0.0472 − 0.998i)19-s + (−1.17 + 2.03i)21-s + (−0.0490 − 0.0849i)23-s + (0.477 + 0.827i)25-s − 2.84·27-s + (0.461 + 0.799i)29-s + 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.936 - 0.350i$
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: \((2,\ 1216,\ (\ :1/2),\ -0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.106893521\)
\(L(\frac12)\) \(\approx\) \(1.106893521\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.206 + 4.35i)T \)
good3 \( 1 + (-1.62 + 2.81i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.235 + 0.407i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.30T + 7T^{2} \)
11 \( 1 + 1.47T + 11T^{2} \)
13 \( 1 + (1.41 + 2.45i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.35 - 5.81i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.235 + 0.407i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.48 - 4.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.74T + 31T^{2} \)
37 \( 1 + 4.74T + 37T^{2} \)
41 \( 1 + (-0.250 + 0.434i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.94 + 3.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.95 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.35 + 4.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.62 + 6.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.26 + 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.316 + 0.548i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.88 + 3.27i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.97 + 3.41i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.41 + 7.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.14T + 83T^{2} \)
89 \( 1 + (1.47 + 2.55i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.0293 - 0.0507i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.975851118617173666788278938713, −8.463040824806342509041252171738, −7.63923961591255830269658555448, −6.66702712141911044040501264093, −6.46339705750386716707840284480, −5.17657214166989618397978478459, −3.51183738828040130706406227892, −2.85132993749347758760712683658, −1.83992054525733159650414014740, −0.38437937399979414730275881956, 2.59662970464139153501344751022, 2.97339254735079609012381496237, 4.17740139947356580544581247697, 4.72681396147408135737281320050, 5.90436160592377187847775757370, 6.89794142546467355075971742395, 7.997948063729415791751874381219, 8.807666685716945763599630177442, 9.623802464000007022092304154669, 9.865577952020762489429156813231

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