Properties

Label 2-1216-152.107-c1-0-14
Degree $2$
Conductor $1216$
Sign $0.998 + 0.0630i$
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  + (−2.71 + 1.56i)3-s + (−1.5 + 0.866i)5-s + 2i·7-s + (3.40 − 5.90i)9-s + (0.225 − 0.391i)13-s + (2.71 − 4.69i)15-s + (−2.90 − 5.03i)17-s + (−4.16 + 1.27i)19-s + (−3.13 − 5.42i)21-s + (−5.70 − 3.29i)23-s + (−1 + 1.73i)25-s + 11.9i·27-s + (2.90 − 5.03i)29-s + 8.33·31-s + (−1.73 − 3i)35-s + ⋯
L(s)  = 1  + (−1.56 + 0.904i)3-s + (−0.670 + 0.387i)5-s + 0.755i·7-s + (1.13 − 1.96i)9-s + (0.0626 − 0.108i)13-s + (0.700 − 1.21i)15-s + (−0.704 − 1.22i)17-s + (−0.956 + 0.292i)19-s + (−0.683 − 1.18i)21-s + (−1.18 − 0.686i)23-s + (−0.200 + 0.346i)25-s + 2.29i·27-s + (0.539 − 0.934i)29-s + 1.49·31-s + (−0.292 − 0.507i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.998 + 0.0630i$
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: \((2,\ 1216,\ (\ :1/2),\ 0.998 + 0.0630i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4890474210\)
\(L(\frac12)\) \(\approx\) \(0.4890474210\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.16 - 1.27i)T \)
good3 \( 1 + (2.71 - 1.56i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-0.225 + 0.391i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.90 + 5.03i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (5.70 + 3.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.90 + 5.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.33T + 31T^{2} \)
37 \( 1 + 3.45T + 37T^{2} \)
41 \( 1 + (0.677 - 0.391i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.276 + 0.478i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.13 - 4.69i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.31 + 7.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.506 + 0.292i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.8 - 6.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.5 + 7.24i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.91 - 8.52i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.18 + 5.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.33T + 83T^{2} \)
89 \( 1 + (9.11 + 5.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.677 - 0.391i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.02294210872463674965850429809, −9.072172525596704239864773229145, −8.149308262097960205727999714114, −6.93649662560297627475192378500, −6.25506653197601154348228824959, −5.51256678773750978783564857619, −4.55235783218757804914923990731, −3.96747922519301113378378953301, −2.53727020270266348833416374544, −0.38148836422816017410721538379, 0.801818992563391391980865125158, 1.99132032170354901986793864279, 4.00948465060556961765980665327, 4.57679640233986540894148321551, 5.70284824981965446110165499149, 6.45964813033117549893488493855, 7.07753015900573235879889416532, 7.968331393613399421408364766006, 8.629264420181450057614579713859, 10.21340868651861116654865878070

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