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Properties

Label 2-1218-7.2-c1-0-13
Degree $2$
Conductor $1218$
Sign $0.779 + 0.626i$
Analytic cond. $9.72577$
Root an. cond. $3.11861$
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)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.956 + 1.65i)5-s − 0.999·6-s + (−2.58 − 0.568i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.956 + 1.65i)10-s + (1.17 + 2.02i)11-s + (−0.499 + 0.866i)12-s + 1.91·13-s + (−1.78 + 1.95i)14-s + 1.91·15-s + (−0.5 + 0.866i)16-s + (2.69 + 4.67i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.427 + 0.740i)5-s − 0.408·6-s + (−0.976 − 0.214i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.302 + 0.523i)10-s + (0.353 + 0.611i)11-s + (−0.144 + 0.249i)12-s + 0.530·13-s + (−0.476 + 0.522i)14-s + 0.493·15-s + (−0.125 + 0.216i)16-s + (0.654 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $0.779 + 0.626i$
Analytic conductor: \(9.72577\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1218} (1045, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1218,\ (\ :1/2),\ 0.779 + 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.431548519\)
\(L(\frac12)\) \(\approx\) \(1.431548519\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.58 + 0.568i)T \)
29 \( 1 - T \)
good5 \( 1 + (0.956 - 1.65i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.17 - 2.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.91T + 13T^{2} \)
17 \( 1 + (-2.69 - 4.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.11 + 5.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.11 + 1.92i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (1.44 + 2.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.612 + 1.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 - 0.313T + 43T^{2} \)
47 \( 1 + (0.0288 - 0.0499i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.56 + 4.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.343 - 0.594i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.14 + 1.97i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.18 - 5.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.37T + 71T^{2} \)
73 \( 1 + (-5.45 - 9.45i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.50 - 9.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.8T + 83T^{2} \)
89 \( 1 + (-1.14 + 1.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.36T + 97T^{2} \)
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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

−9.774637202448881400450742412212, −9.049421327644493190073161416172, −7.83659205883531575138209341119, −6.98860050425827533436915164546, −6.39678650488688520086532304240, −5.47558545641340903426984084218, −4.19355345875756338261011004645, −3.39637074209298066839200436131, −2.46118953219168902392053807676, −0.949377276725421672081466112012, 0.801332111557492907798730196116, 3.09893361553798881765003525842, 3.75611600013567359986008124585, 4.77817236986403485277904828200, 5.67776422743676887675907200667, 6.22630289427648993938721932314, 7.34079337967561079389691947536, 8.159525868978731236788746666430, 9.106906688647096000639811215548, 9.514304650796730200389921259347

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