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Properties

Label 2-242-11.5-c1-0-5
Degree $2$
Conductor $242$
Sign $-0.0783 + 0.996i$
Analytic cond. $1.93237$
Root an. cond. $1.39010$
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  + (−0.309 − 0.951i)2-s + (−0.592 − 0.430i)3-s + (−0.809 + 0.587i)4-s + (−0.535 + 1.64i)5-s + (−0.226 + 0.696i)6-s + (3.82 − 2.78i)7-s + (0.809 + 0.587i)8-s + (−0.761 − 2.34i)9-s + 1.73·10-s + 0.732·12-s + (−0.927 − 2.85i)13-s + (−3.82 − 2.78i)14-s + (1.02 − 0.745i)15-s + (0.309 − 0.951i)16-s + (1.60 − 4.94i)17-s + (−1.99 + 1.44i)18-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.341 − 0.248i)3-s + (−0.404 + 0.293i)4-s + (−0.239 + 0.736i)5-s + (−0.0923 + 0.284i)6-s + (1.44 − 1.05i)7-s + (0.286 + 0.207i)8-s + (−0.253 − 0.781i)9-s + 0.547·10-s + 0.211·12-s + (−0.257 − 0.791i)13-s + (−1.02 − 0.743i)14-s + (0.264 − 0.192i)15-s + (0.0772 − 0.237i)16-s + (0.389 − 1.19i)17-s + (−0.469 + 0.341i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0783 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0783 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $-0.0783 + 0.996i$
Analytic conductor: \(1.93237\)
Root analytic conductor: \(1.39010\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{242} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 242,\ (\ :1/2),\ -0.0783 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678802 - 0.734265i\)
\(L(\frac12)\) \(\approx\) \(0.678802 - 0.734265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good3 \( 1 + (0.592 + 0.430i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.535 - 1.64i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-3.82 + 2.78i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.927 + 2.85i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.60 + 4.94i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.02 - 0.745i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.26T + 23T^{2} \)
29 \( 1 + (2.42 - 1.76i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.0606 - 0.186i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.82 + 4.22i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.650 + 0.472i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-6.63 - 4.81i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.45 - 10.6i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.65 - 5.56i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.783 - 2.41i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + (2.92 - 9.00i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.750 - 0.545i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.46 - 4.50i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.53 - 7.79i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 6.46T + 89T^{2} \)
97 \( 1 + (0.309 + 0.951i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.61960352124192003759012324783, −11.02715704917790115045661618500, −10.27835187639088217070579456424, −9.066358542982056099144250277653, −7.69436890189288659943328642939, −7.23680764751147333042318409073, −5.56294786868471239030670604670, −4.26868282590240362558673437447, −2.96635171083309175710332525999, −1.02054133062628186279778472957, 1.89003009872714159174122067251, 4.45195324283088068808126145144, 5.13386039490432478840979664405, 6.06372241257703946075242763506, 7.75320363200552935131872194897, 8.348659797670763256090687180223, 9.148298352906073614789668973835, 10.47372821948765557451304786321, 11.49809378103700360964807445659, 12.15966617283517753671347124021

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