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Properties

Label 2-1218-203.144-c1-0-4
Degree $2$
Conductor $1218$
Sign $0.317 - 0.948i$
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 + (1.74 + 3.02i)5-s − 0.999·6-s + (−0.738 + 2.54i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (3.02 + 1.74i)10-s + (−0.0483 − 0.0279i)11-s + (−0.866 + 0.499i)12-s + 1.05·13-s + (0.630 + 2.56i)14-s − 3.49i·15-s + (−0.5 − 0.866i)16-s + (−2.24 − 1.29i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.781 + 1.35i)5-s − 0.408·6-s + (−0.279 + 0.960i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.957 + 0.552i)10-s + (−0.0145 − 0.00842i)11-s + (−0.249 + 0.144i)12-s + 0.293·13-s + (0.168 + 0.686i)14-s − 0.902i·15-s + (−0.125 − 0.216i)16-s + (−0.544 − 0.314i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $0.317 - 0.948i$
Analytic conductor: \(9.72577\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1218} (1159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1218,\ (\ :1/2),\ 0.317 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.847462324\)
\(L(\frac12)\) \(\approx\) \(1.847462324\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.738 - 2.54i)T \)
29 \( 1 + (4.72 + 2.58i)T \)
good5 \( 1 + (-1.74 - 3.02i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.0483 + 0.0279i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.05T + 13T^{2} \)
17 \( 1 + (2.24 + 1.29i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.57 - 3.79i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.11 - 5.39i)T + (-11.5 + 19.9i)T^{2} \)
31 \( 1 + (-5.37 - 3.10i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.15 + 1.82i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.82iT - 41T^{2} \)
43 \( 1 - 0.955iT - 43T^{2} \)
47 \( 1 + (8.66 - 5.00i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.37 + 9.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.84 - 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.33 + 1.34i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.06 - 3.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.17T + 71T^{2} \)
73 \( 1 + (-12.3 - 7.11i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.13 + 2.96i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + (-15.3 + 8.87i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.61iT - 97T^{2} \)
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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

−10.02796015273522087842213452853, −9.397300494715544400997968872034, −8.209260487762056128040053338807, −7.02559400621563316415827918507, −6.26627986252431191933798583397, −5.95628244133394873654758208929, −4.91112335172412656399505238761, −3.57227945173278078457073744887, −2.59976522163922193945158783039, −1.83002027990250359351594739183, 0.65716127204494525882858523403, 2.16153944331414212310156568321, 3.80192844450876552062801269206, 4.62164732325156309116762093506, 5.09712719133856969055295518570, 6.32207926217776347442953914266, 6.62066618344779909296007300802, 7.961250373998355052385901178470, 8.822803736640542887248740171958, 9.445628589110513037713999476216

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