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Properties

Label 2-1216-152.107-c1-0-21
Degree $2$
Conductor $1216$
Sign $0.940 - 0.339i$
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  + (1.22 − 0.709i)3-s + (2.34 − 1.35i)5-s + 4.52i·7-s + (−0.492 + 0.853i)9-s − 4.04·11-s + (1.51 − 2.63i)13-s + (1.92 − 3.32i)15-s + (2.88 + 4.98i)17-s + (4.10 + 1.45i)19-s + (3.21 + 5.56i)21-s + (1.92 + 1.10i)23-s + (1.16 − 2.01i)25-s + 5.65i·27-s + (3.96 − 6.86i)29-s + 7.84·31-s + ⋯
L(s)  = 1  + (0.709 − 0.409i)3-s + (1.04 − 0.605i)5-s + 1.71i·7-s + (−0.164 + 0.284i)9-s − 1.21·11-s + (0.421 − 0.729i)13-s + (0.495 − 0.858i)15-s + (0.698 + 1.20i)17-s + (0.942 + 0.334i)19-s + (0.701 + 1.21i)21-s + (0.400 + 0.231i)23-s + (0.232 − 0.402i)25-s + 1.08i·27-s + (0.735 − 1.27i)29-s + 1.40·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.940 - 0.339i$
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: \((2,\ 1216,\ (\ :1/2),\ 0.940 - 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.431624901\)
\(L(\frac12)\) \(\approx\) \(2.431624901\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.10 - 1.45i)T \)
good3 \( 1 + (-1.22 + 0.709i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.34 + 1.35i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 4.52iT - 7T^{2} \)
11 \( 1 + 4.04T + 11T^{2} \)
13 \( 1 + (-1.51 + 2.63i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.88 - 4.98i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.92 - 1.10i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.96 + 6.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.84T + 31T^{2} \)
37 \( 1 + 6.02T + 37T^{2} \)
41 \( 1 + (-2.97 + 1.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.63 - 4.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.05 + 4.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.93 + 5.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.972 - 0.561i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.77 - 5.64i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.69 + 5.01i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.269 + 0.466i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.22 + 5.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.40 - 4.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + (14.5 + 8.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.0 + 6.97i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.785709374857624107147748938711, −8.651068273854892429037484944000, −8.390641300550333159240451495251, −7.66409599467052179189064813218, −6.09154841012651964248227522915, −5.60439684037151212016002573801, −5.03097498001627351625348602832, −3.14258924874976619096177253132, −2.47762958254823272990936311110, −1.54727818777250668562775320643, 1.04437721807334096919132155745, 2.70845396016249403588901722893, 3.26491320279605389777670125640, 4.45014247000898562971166380822, 5.36123569411838985458261460130, 6.58884966735001336570978038595, 7.14933430943231496185823167868, 8.030553068243730589109062830016, 9.051330256079251499759153557372, 9.882330373987438493936387668358

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