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Properties

Label 2-1216-19.7-c1-0-18
Degree $2$
Conductor $1216$
Sign $0.813 + 0.582i$
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  + (−0.5 − 0.866i)3-s + 4·7-s + (1 − 1.73i)9-s + 3·11-s + (1 − 1.73i)13-s + (3 + 5.19i)17-s + (−3.5 + 2.59i)19-s + (−2 − 3.46i)21-s + (−3 + 5.19i)23-s + (2.5 − 4.33i)25-s − 5·27-s − 2·31-s + (−1.5 − 2.59i)33-s + 10·37-s − 1.99·39-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 1.51·7-s + (0.333 − 0.577i)9-s + 0.904·11-s + (0.277 − 0.480i)13-s + (0.727 + 1.26i)17-s + (−0.802 + 0.596i)19-s + (−0.436 − 0.755i)21-s + (−0.625 + 1.08i)23-s + (0.5 − 0.866i)25-s − 0.962·27-s − 0.359·31-s + (−0.261 − 0.452i)33-s + 1.64·37-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.813 + 0.582i$
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: \((2,\ 1216,\ (\ :1/2),\ 0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.960454982\)
\(L(\frac12)\) \(\approx\) \(1.960454982\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 2.59i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.702053998763435779040645437067, −8.594997238356258866197640668077, −8.056546432644368456704969429313, −7.25598324195587839297105434044, −6.18713490506422679275455789007, −5.65996942841242728050958522988, −4.35404747492204119635745182003, −3.69415207664286162707402443037, −1.90330062606137141653776220920, −1.14937865787251784627735637386, 1.26275722374959199203118865377, 2.42178249573288638223281124557, 4.02437107510888914224771831066, 4.67106524340031421225849471868, 5.32444225123236410546874657033, 6.50431025541065414667769076244, 7.42396357932175895233553649356, 8.195864169652111299420725846120, 9.031136496353955283587810119219, 9.816205521790821191649931447627

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