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Properties

Label 2-1216-76.31-c1-0-7
Degree $2$
Conductor $1216$
Sign $-0.417 - 0.908i$
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  + (0.637 + 1.10i)3-s + (1.60 + 2.77i)5-s + 1.25i·7-s + (0.688 − 1.19i)9-s + 2.11i·11-s + (−2.12 − 1.22i)13-s + (−2.04 + 3.53i)15-s + (0.765 + 1.32i)17-s + (−3.76 + 2.19i)19-s + (−1.37 + 0.796i)21-s + (7.61 + 4.39i)23-s + (−2.64 + 4.57i)25-s + 5.57·27-s + (5.20 + 3.00i)29-s − 7.78·31-s + ⋯
L(s)  = 1  + (0.367 + 0.637i)3-s + (0.717 + 1.24i)5-s + 0.472i·7-s + (0.229 − 0.397i)9-s + 0.636i·11-s + (−0.590 − 0.341i)13-s + (−0.527 + 0.913i)15-s + (0.185 + 0.321i)17-s + (−0.863 + 0.504i)19-s + (−0.301 + 0.173i)21-s + (1.58 + 0.917i)23-s + (−0.528 + 0.914i)25-s + 1.07·27-s + (0.967 + 0.558i)29-s − 1.39·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.417 - 0.908i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.417 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.024725675\)
\(L(\frac12)\) \(\approx\) \(2.024725675\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.76 - 2.19i)T \)
good3 \( 1 + (-0.637 - 1.10i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.60 - 2.77i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.25iT - 7T^{2} \)
11 \( 1 - 2.11iT - 11T^{2} \)
13 \( 1 + (2.12 + 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.765 - 1.32i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-7.61 - 4.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.20 - 3.00i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.78T + 31T^{2} \)
37 \( 1 + 9.97iT - 37T^{2} \)
41 \( 1 + (-1.09 + 0.631i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.04 - 2.91i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.12 + 3.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.18 + 3.57i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.83 + 4.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.80 - 4.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0235 - 0.0408i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.12 - 5.41i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.658 - 1.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.77 - 6.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.84iT - 83T^{2} \)
89 \( 1 + (6.02 + 3.47i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.51 + 4.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.930297564976434082742544978558, −9.396250486259978805584418989875, −8.572156971654281768047756971871, −7.32004513085687527246207507840, −6.77691428508652248315426730802, −5.80904243089063513602267832452, −4.89622299182986509117394761913, −3.66993057422961581593509505151, −2.91037463125627058054135140436, −1.87798175826145707021823497522, 0.839731580083763358103983801992, 1.87957751273009574078253571767, 2.98714430545485521610856607887, 4.65015965061072872611120592445, 4.93922424151873473921896351062, 6.24980003929915021151682406471, 7.00732368273116638523484716669, 7.941962984474611954283387533436, 8.703962697591025673673070488733, 9.237958454183641360794270438557

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