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Properties

Label 2-29e2-29.20-c1-0-36
Degree $2$
Conductor $841$
Sign $0.973 - 0.230i$
Analytic cond. $6.71541$
Root an. cond. $2.59141$
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  + (1.45 − 0.702i)2-s + (1.00 + 1.26i)3-s + (0.385 − 0.483i)4-s + (2.57 − 1.23i)5-s + (2.35 + 1.13i)6-s + (1.39 + 1.74i)7-s + (−0.497 + 2.18i)8-s + (0.0849 − 0.372i)9-s + (2.87 − 3.61i)10-s + (0.805 + 3.52i)11-s + 1.00·12-s + (−0.942 − 4.12i)13-s + (3.25 + 1.56i)14-s + (4.16 + 2.00i)15-s + (1.08 + 4.73i)16-s − 6.61·17-s + ⋯
L(s)  = 1  + (1.03 − 0.496i)2-s + (0.582 + 0.730i)3-s + (0.192 − 0.241i)4-s + (1.14 − 0.553i)5-s + (0.962 + 0.463i)6-s + (0.526 + 0.660i)7-s + (−0.175 + 0.770i)8-s + (0.0283 − 0.124i)9-s + (0.910 − 1.14i)10-s + (0.242 + 1.06i)11-s + 0.288·12-s + (−0.261 − 1.14i)13-s + (0.871 + 0.419i)14-s + (1.07 + 0.517i)15-s + (0.270 + 1.18i)16-s − 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.973 - 0.230i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.973 - 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.68085 + 0.430009i\)
\(L(\frac12)\) \(\approx\) \(3.68085 + 0.430009i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (-1.45 + 0.702i)T + (1.24 - 1.56i)T^{2} \)
3 \( 1 + (-1.00 - 1.26i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-2.57 + 1.23i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (-1.39 - 1.74i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-0.805 - 3.52i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.942 + 4.12i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 6.61T + 17T^{2} \)
19 \( 1 + (-1.15 + 1.44i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (2.91 + 1.40i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (0.982 - 0.473i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (-1.93 + 8.48i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + 2.85T + 41T^{2} \)
43 \( 1 + (-2.49 - 1.19i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.55 - 6.82i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-1.80 + 0.867i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + (-1.00 - 1.26i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-2.33 + 10.2i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (0.339 + 1.48i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.262 - 0.126i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (1.13 - 4.96i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (-4.95 + 6.21i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-7.84 + 3.77i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (10.3 - 12.9i)T + (-21.5 - 94.5i)T^{2} \)
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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

−10.22215811895660845863753220056, −9.258083758603875122293667234329, −8.945320101486992317982233128049, −7.890300201952330538250133311499, −6.38634295075466272444854852062, −5.39902647346666448941022504883, −4.77335719026036043923632264832, −3.99378587790840757776929968844, −2.66986137465190042085463418423, −1.99854907176176014423122560676, 1.54204202526860420064606589520, 2.59300649132896447031652840290, 3.91544440950170006030493650039, 4.84842231810278988745550621505, 5.94849435844127955424670144274, 6.63667759969722164238508978767, 7.20611933335287265483362102261, 8.351066315442525659315159930979, 9.259048710602945566915933062070, 10.16720334699169600597403147759

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