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Properties

Label 2-1242-9.7-c1-0-6
Degree $2$
Conductor $1242$
Sign $0.325 - 0.945i$
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 + (−0.536 − 0.928i)5-s + (−0.121 + 0.211i)7-s + 0.999·8-s + 1.07·10-s + (0.317 − 0.549i)11-s + (2.75 + 4.77i)13-s + (−0.121 − 0.211i)14-s + (−0.5 + 0.866i)16-s − 7.28·17-s + 1.65·19-s + (−0.536 + 0.928i)20-s + (0.317 + 0.549i)22-s + (0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.239 − 0.415i)5-s + (−0.0460 + 0.0798i)7-s + 0.353·8-s + 0.339·10-s + (0.0956 − 0.165i)11-s + (0.764 + 1.32i)13-s + (−0.0325 − 0.0564i)14-s + (−0.125 + 0.216i)16-s − 1.76·17-s + 0.379·19-s + (−0.119 + 0.207i)20-s + (0.0676 + 0.117i)22-s + (0.104 + 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1242\)    =    \(2 \cdot 3^{3} \cdot 23\)
Sign: $0.325 - 0.945i$
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: \((2,\ 1242,\ (\ :1/2),\ 0.325 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.171160154\)
\(L(\frac12)\) \(\approx\) \(1.171160154\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.536 + 0.928i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.121 - 0.211i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.317 + 0.549i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.75 - 4.77i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.28T + 17T^{2} \)
19 \( 1 - 1.65T + 19T^{2} \)
29 \( 1 + (-2.59 + 4.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.82 - 6.63i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.11T + 37T^{2} \)
41 \( 1 + (-3.51 - 6.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.73 + 4.72i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.39 - 4.15i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.94T + 53T^{2} \)
59 \( 1 + (-5.47 - 9.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.02 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.121 + 0.211i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.03T + 71T^{2} \)
73 \( 1 + 0.0957T + 73T^{2} \)
79 \( 1 + (6.05 - 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.66 + 8.08i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.00T + 89T^{2} \)
97 \( 1 + (-4.59 + 7.95i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.609724201756788166605158035213, −8.751594967568930302562685562173, −8.561382877894653552073986330776, −7.33836745869350871432807105781, −6.57934575776767530799088091987, −5.93610717472353082255950451107, −4.60808137707408103449433150665, −4.17301859948698313684499985870, −2.52273095373550513675993846150, −1.09029803298247973131339811537, 0.69330760754172318773167632367, 2.25174261159652806885922666383, 3.22565980988562059487186633053, 4.12319395208896675037359953127, 5.20400679739165366893600602271, 6.34421397770029050050197836422, 7.17455298217994493133627951100, 8.082279684762450625719476945102, 8.771637659227669382417441962940, 9.598555102141238824215883996273

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