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Properties

Label 2-1216-152.45-c1-0-27
Degree $2$
Conductor $1216$
Sign $-0.547 + 0.837i$
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  + (−0.495 + 0.285i)3-s + (−0.355 + 0.205i)5-s + 0.565·7-s + (−1.33 + 2.31i)9-s + 0.623i·11-s + (−3.87 − 2.23i)13-s + (0.117 − 0.203i)15-s + (−1.03 − 1.79i)17-s + (−1.77 − 3.98i)19-s + (−0.280 + 0.161i)21-s + (2.39 − 4.15i)23-s + (−2.41 + 4.18i)25-s − 3.24i·27-s + (−2.28 − 1.31i)29-s + 5.12·31-s + ⋯
L(s)  = 1  + (−0.285 + 0.165i)3-s + (−0.159 + 0.0918i)5-s + 0.213·7-s + (−0.445 + 0.771i)9-s + 0.187i·11-s + (−1.07 − 0.620i)13-s + (0.0303 − 0.0525i)15-s + (−0.251 − 0.434i)17-s + (−0.407 − 0.913i)19-s + (−0.0611 + 0.0352i)21-s + (0.500 − 0.866i)23-s + (−0.483 + 0.836i)25-s − 0.624i·27-s + (−0.424 − 0.245i)29-s + 0.920·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.547 + 0.837i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.547 + 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4629437585\)
\(L(\frac12)\) \(\approx\) \(0.4629437585\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.77 + 3.98i)T \)
good3 \( 1 + (0.495 - 0.285i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.355 - 0.205i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.565T + 7T^{2} \)
11 \( 1 - 0.623iT - 11T^{2} \)
13 \( 1 + (3.87 + 2.23i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.03 + 1.79i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.39 + 4.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.28 + 1.31i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.12T + 31T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + (-1.11 - 1.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.626 - 0.361i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.18 + 7.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.5 + 6.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.71 - 4.45i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0512 + 0.0295i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.447 + 0.258i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.32 + 7.49i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.86 - 4.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.50 + 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2iT - 83T^{2} \)
89 \( 1 + (4.25 - 7.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.33 + 9.24i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.516695962655847064101417977707, −8.616530570908869619363702132023, −7.75014752234698578663613654951, −7.10360070707464787762148296389, −6.01988184889121066900210147826, −5.01259860471686596352689122269, −4.56195287719976858921246506538, −3.05254204900113385565160602322, −2.16087409356874628013293107409, −0.19932448122006334880843094044, 1.49055881634252613735847919582, 2.84812228293723271453788047590, 3.98390164385894592537363255414, 4.88658681619848144740843996236, 5.95861598107123759607531402331, 6.57727203113232813417307691956, 7.59341465234989238101890048241, 8.337494428706920100963769065164, 9.264564052437755466354940406532, 9.894545259465499819367363999420

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