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Properties

Label 2-1216-152.125-c1-0-38
Degree $2$
Conductor $1216$
Sign $-0.659 - 0.751i$
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  + (−1.80 − 1.04i)3-s + (−0.410 − 0.236i)5-s − 3.88·7-s + (0.675 + 1.17i)9-s − 6.19i·11-s + (4.97 − 2.87i)13-s + (0.494 + 0.856i)15-s + (0.442 − 0.766i)17-s + (4.19 + 1.19i)19-s + (7.01 + 4.04i)21-s + (−0.917 − 1.58i)23-s + (−2.38 − 4.13i)25-s + 3.43i·27-s + (−6.58 + 3.79i)29-s − 6.70·31-s + ⋯
L(s)  = 1  + (−1.04 − 0.602i)3-s + (−0.183 − 0.105i)5-s − 1.46·7-s + (0.225 + 0.390i)9-s − 1.86i·11-s + (1.38 − 0.797i)13-s + (0.127 + 0.221i)15-s + (0.107 − 0.185i)17-s + (0.961 + 0.274i)19-s + (1.53 + 0.883i)21-s + (−0.191 − 0.331i)23-s + (−0.477 − 0.827i)25-s + 0.661i·27-s + (−1.22 + 0.705i)29-s − 1.20·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.659 - 0.751i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2260667557\)
\(L(\frac12)\) \(\approx\) \(0.2260667557\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.19 - 1.19i)T \)
good3 \( 1 + (1.80 + 1.04i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.410 + 0.236i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.88T + 7T^{2} \)
11 \( 1 + 6.19iT - 11T^{2} \)
13 \( 1 + (-4.97 + 2.87i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.442 + 0.766i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.917 + 1.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.58 - 3.79i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.70T + 31T^{2} \)
37 \( 1 - 1.54iT - 37T^{2} \)
41 \( 1 + (3.44 - 5.96i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.45 + 3.72i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.71 - 11.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.06 + 1.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.1 + 5.88i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.12 + 1.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.74 + 5.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.98 - 6.89i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.01 - 5.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.52 - 4.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (1.84 + 3.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.213 - 0.369i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.163461757188415863633263415883, −8.412274908705067098759578813093, −7.43670964286751721353054997370, −6.37106227015260246515728540338, −5.99548955843686254599990737275, −5.41174755015018353689122576703, −3.57368422292767722517404593643, −3.22563532855132568011609672387, −1.09544008456051755873350112503, −0.12985828564977278660596589226, 1.88219687898259032295229213789, 3.54907883559732522780167313875, 4.12543811274295623126316359859, 5.31433918312280420603681937752, 5.97152289670517440814493306726, 6.90461976585277841838210167162, 7.46473028129161998281899969811, 8.988602807826390334644657629215, 9.623350496942387404172146162413, 10.14009577676864575306482236929

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