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Properties

Label 2-1218-203.144-c1-0-24
Degree $2$
Conductor $1218$
Sign $0.667 + 0.744i$
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 + (−1.47 − 2.55i)5-s − 0.999·6-s + (1.71 − 2.01i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (2.55 + 1.47i)10-s + (1.36 + 0.789i)11-s + (0.866 − 0.499i)12-s + 5.63·13-s + (−0.483 + 2.60i)14-s − 2.94i·15-s + (−0.5 − 0.866i)16-s + (0.231 + 0.133i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.659 − 1.14i)5-s − 0.408·6-s + (0.649 − 0.760i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.807 + 0.466i)10-s + (0.412 + 0.237i)11-s + (0.249 − 0.144i)12-s + 1.56·13-s + (−0.129 + 0.695i)14-s − 0.761i·15-s + (−0.125 − 0.216i)16-s + (0.0561 + 0.0324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $0.667 + 0.744i$
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: \((2,\ 1218,\ (\ :1/2),\ 0.667 + 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.418773603\)
\(L(\frac12)\) \(\approx\) \(1.418773603\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 2.01i)T \)
29 \( 1 + (3.00 - 4.46i)T \)
good5 \( 1 + (1.47 + 2.55i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.36 - 0.789i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.63T + 13T^{2} \)
17 \( 1 + (-0.231 - 0.133i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.24 - 1.87i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.48 + 4.30i)T + (-11.5 + 19.9i)T^{2} \)
31 \( 1 + (-1.67 - 0.965i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.44 + 4.30i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 + 7.42iT - 43T^{2} \)
47 \( 1 + (-3.86 + 2.23i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.43 - 4.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.170 + 0.295i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.48 - 3.16i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.86 + 6.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.87T + 71T^{2} \)
73 \( 1 + (-1.52 - 0.881i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.934 + 0.539i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + (3.51 - 2.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.3iT - 97T^{2} \)
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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

−9.247140081272191990125452064609, −8.714948620974787024303072195702, −8.180895825637971750139310347302, −7.48568214532705451542775925492, −6.43062851664222737570993430653, −5.33166625175942494451577202046, −4.23856779271493875402837966637, −3.83044058794152674627881100743, −1.87762960768560108355046098502, −0.790301663433346353378045439772, 1.39188587415440146880774297593, 2.58453132801302912091072755632, 3.41109477198120986170169121835, 4.32270793770606450292125082399, 6.05432502522391240486796004832, 6.55299086812146822423797887618, 7.83009038495105652823186153509, 8.047596243125692608771268755929, 8.969371769500695207612146592460, 9.720689167562130318942624683399

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