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Properties

Label 2-1242-9.7-c1-0-12
Degree $2$
Conductor $1242$
Sign $0.993 + 0.111i$
Analytic cond. $9.91741$
Root an. cond. $3.14919$
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 + (−0.499 − 0.866i)4-s + (−0.217 − 0.377i)5-s + (2.31 − 4.00i)7-s + 0.999·8-s + 0.435·10-s + (−2.10 + 3.63i)11-s + (1.60 + 2.77i)13-s + (2.31 + 4.00i)14-s + (−0.5 + 0.866i)16-s + 5.97·17-s − 6.45·19-s + (−0.217 + 0.377i)20-s + (−2.10 − 3.63i)22-s + (0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.0973 − 0.168i)5-s + (0.874 − 1.51i)7-s + 0.353·8-s + 0.137·10-s + (−0.633 + 1.09i)11-s + (0.444 + 0.769i)13-s + (0.618 + 1.07i)14-s + (−0.125 + 0.216i)16-s + 1.44·17-s − 1.48·19-s + (−0.0486 + 0.0843i)20-s + (−0.447 − 0.775i)22-s + (0.104 + 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1242\)    =    \(2 \cdot 3^{3} \cdot 23\)
Sign: $0.993 + 0.111i$
Analytic conductor: \(9.91741\)
Root analytic conductor: \(3.14919\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1242} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1242,\ (\ :1/2),\ 0.993 + 0.111i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.417720976\)
\(L(\frac12)\) \(\approx\) \(1.417720976\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.217 + 0.377i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.31 + 4.00i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.10 - 3.63i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.60 - 2.77i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.97T + 17T^{2} \)
19 \( 1 + 6.45T + 19T^{2} \)
29 \( 1 + (-1.77 + 3.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.03 - 3.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.64T + 37T^{2} \)
41 \( 1 + (1.97 + 3.42i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.28 + 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.71 + 8.17i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.713T + 53T^{2} \)
59 \( 1 + (-1.06 - 1.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.20 + 7.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.31 - 4.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.53T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + (3.42 - 5.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.57 - 6.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + (-3.77 + 6.53i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.826010838912793658973277269903, −8.602034809111633733573788256444, −8.023272609555306497827504051985, −7.26021033790343101229381549438, −6.68635067769182677688546474195, −5.44720418118504080975015213391, −4.49208080324274801689748876877, −3.99127210302341651485307199678, −2.08838721148899656832733481514, −0.843562838153737774318966616727, 1.15112008761602440674510232431, 2.56912783823698972617323174867, 3.15239848904030975788836759723, 4.56543286251445437971215650146, 5.59255138725250919345350840420, 6.10348723124604186537046538158, 7.75597434262264483866401963340, 8.208897488949506526603046721181, 8.809536382941306991534882212140, 9.699895715318370329549053844533

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