Properties

Label 2-1216-152.107-c1-0-10
Degree $2$
Conductor $1216$
Sign $-0.114 - 0.993i$
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  + (−2.36 + 1.36i)3-s + (−3.63 + 2.09i)5-s − 0.517i·7-s + (2.22 − 3.86i)9-s + 5.15·11-s + (1.67 − 2.89i)13-s + (5.72 − 9.91i)15-s + (−0.818 − 1.41i)17-s + (3.66 + 2.36i)19-s + (0.706 + 1.22i)21-s + (5.72 + 3.30i)23-s + (6.29 − 10.9i)25-s + 3.98i·27-s + (−3.01 + 5.23i)29-s − 0.896·31-s + ⋯
L(s)  = 1  + (−1.36 + 0.788i)3-s + (−1.62 + 0.937i)5-s − 0.195i·7-s + (0.743 − 1.28i)9-s + 1.55·11-s + (0.463 − 0.803i)13-s + (1.47 − 2.56i)15-s + (−0.198 − 0.343i)17-s + (0.840 + 0.541i)19-s + (0.154 + 0.267i)21-s + (1.19 + 0.689i)23-s + (1.25 − 2.18i)25-s + 0.766i·27-s + (−0.560 + 0.971i)29-s − 0.161·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.114 - 0.993i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.114 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7192986916\)
\(L(\frac12)\) \(\approx\) \(0.7192986916\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.66 - 2.36i)T \)
good3 \( 1 + (2.36 - 1.36i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (3.63 - 2.09i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.517iT - 7T^{2} \)
11 \( 1 - 5.15T + 11T^{2} \)
13 \( 1 + (-1.67 + 2.89i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.818 + 1.41i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-5.72 - 3.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.01 - 5.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.896T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 + (5.18 - 2.99i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.89 + 5.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.89 - 1.67i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.40 + 7.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.109 + 0.0629i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.1 + 5.86i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.6 - 6.13i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.44 - 7.69i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.88 - 4.99i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.712 + 1.23i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.864T + 83T^{2} \)
89 \( 1 + (-12.8 - 7.40i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.0530 + 0.0306i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.27153536644148324057959578459, −9.285803541617236452276259523232, −8.283885372139778017029628390953, −7.09846801102442232648945873016, −6.83924451251231023833809221679, −5.66819050161931362194168217151, −4.80436791499156493404974363220, −3.64529425939957986609749910438, −3.51232371306706311871248471663, −0.887488394968239605266593126570, 0.58631102097798631622268606752, 1.48163247462308248258028728729, 3.62428957058066568439136012028, 4.44504446655027243017072706612, 5.20985189313367355900402301822, 6.32821771294617187954298577112, 6.97369267127228143765729406563, 7.65520406596562241204126717789, 8.803984886974271385883789550531, 9.148982245799757899655535416060

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