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Properties

Label 2-1216-19.7-c1-0-28
Degree $2$
Conductor $1216$
Sign $-0.444 + 0.895i$
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  + (−0.684 − 1.18i)3-s + (0.780 + 1.35i)5-s + (0.561 − 0.972i)9-s − 3.50·11-s + (−0.219 + 0.379i)13-s + (1.06 − 1.85i)15-s + (0.219 + 0.379i)17-s + (−3.12 − 3.03i)19-s + (2.43 − 4.22i)23-s + (1.28 − 2.21i)25-s − 5.64·27-s + (2.34 − 4.05i)29-s + 2.73·31-s + (2.40 + 4.16i)33-s + 1.12·37-s + ⋯
L(s)  = 1  + (−0.395 − 0.684i)3-s + (0.349 + 0.604i)5-s + (0.187 − 0.324i)9-s − 1.05·11-s + (−0.0608 + 0.105i)13-s + (0.276 − 0.478i)15-s + (0.0531 + 0.0920i)17-s + (−0.716 − 0.697i)19-s + (0.508 − 0.881i)23-s + (0.256 − 0.443i)25-s − 1.08·27-s + (0.434 − 0.753i)29-s + 0.492·31-s + (0.418 + 0.724i)33-s + 0.184·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.038436395\)
\(L(\frac12)\) \(\approx\) \(1.038436395\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.12 + 3.03i)T \)
good3 \( 1 + (0.684 + 1.18i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.780 - 1.35i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 3.50T + 11T^{2} \)
13 \( 1 + (0.219 - 0.379i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.219 - 0.379i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.43 + 4.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.34 + 4.05i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.73T + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.80 + 6.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.06 + 1.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.219 + 0.379i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.684 - 1.18i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.21 - 2.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.93 + 12.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.43 + 4.22i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.623 + 1.07i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.06 - 1.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.24T + 83T^{2} \)
89 \( 1 + (-5.34 + 9.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.62 + 9.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.578931395764229975034379153810, −8.536968828504466334834349700124, −7.73820114006778916792046388884, −6.71061306237431680381966883149, −6.45141197077749392455772186156, −5.34429625952117872511036583141, −4.34881978515194701260541971975, −2.96315952666570897498502079853, −2.08264371981845004653990417402, −0.46435937035163675947438230143, 1.49161751235416593924183436488, 2.86199543812646480683781857797, 4.11166258865235076864890157345, 5.10103508421786210570398589015, 5.39983451951945533811726287281, 6.57769279753598816988349611444, 7.70266043540801210118422425125, 8.351815449465270036728606736982, 9.388591931228191845755850688731, 9.994925995849952049432431944428

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