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Properties

Label 2-1218-203.86-c1-0-29
Degree $2$
Conductor $1218$
Sign $0.995 - 0.0978i$
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  + (0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (1.83 − 3.17i)5-s − 0.999·6-s + (2.42 + 1.06i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (3.17 − 1.83i)10-s + (1.01 − 0.584i)11-s + (−0.866 − 0.499i)12-s − 1.90·13-s + (1.56 + 2.13i)14-s + 3.67i·15-s + (−0.5 + 0.866i)16-s + (−2.60 + 1.50i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.821 − 1.42i)5-s − 0.408·6-s + (0.915 + 0.403i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (1.00 − 0.580i)10-s + (0.305 − 0.176i)11-s + (−0.249 − 0.144i)12-s − 0.529·13-s + (0.417 + 0.570i)14-s + 0.948i·15-s + (−0.125 + 0.216i)16-s + (−0.631 + 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $0.995 - 0.0978i$
Analytic conductor: \(9.72577\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1218} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1218,\ (\ :1/2),\ 0.995 - 0.0978i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.591581414\)
\(L(\frac12)\) \(\approx\) \(2.591581414\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-2.42 - 1.06i)T \)
29 \( 1 + (-3.02 + 4.45i)T \)
good5 \( 1 + (-1.83 + 3.17i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.01 + 0.584i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.90T + 13T^{2} \)
17 \( 1 + (2.60 - 1.50i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.68 - 3.28i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.47 + 6.02i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (3.61 - 2.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.88 - 1.66i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.74iT - 41T^{2} \)
43 \( 1 + 6.27iT - 43T^{2} \)
47 \( 1 + (1.49 + 0.862i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.26 + 7.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.49 - 9.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.07 - 5.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.07 - 7.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.03T + 71T^{2} \)
73 \( 1 + (8.12 - 4.69i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.60 + 0.924i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.81T + 83T^{2} \)
89 \( 1 + (-7.21 - 4.16i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.93iT - 97T^{2} \)
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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

−9.668501291814932598560729500320, −8.787371566031553394559845462291, −8.303510800426690686101021515977, −7.12906674755911443005452181560, −6.04192655531549369358632267619, −5.36579917627907953771320987813, −4.85777895956149997063427216970, −4.05112636471626302126586126726, −2.37638251155815831673414590631, −1.16452892251125479671257908615, 1.38738688531366645154078979598, 2.45161352697304177860239862796, 3.38530809510383403832080700678, 4.76300255635940147191588984013, 5.38247066119020395175653431151, 6.40823459044279329708724818517, 7.11562447148227417192182842401, 7.60950088480598893732717569850, 9.273597690119308798002003593802, 9.877793645005112221625364725882

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