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Properties

Label 2-29e2-29.4-c1-0-40
Degree $2$
Conductor $841$
Sign $-0.317 + 0.948i$
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  + (1.57 + 0.360i)2-s + (−0.702 − 1.45i)3-s + (0.556 + 0.268i)4-s + (−0.635 + 2.78i)5-s + (−0.582 − 2.55i)6-s + (−2.01 + 0.970i)7-s + (−1.74 − 1.39i)8-s + (0.238 − 0.298i)9-s + (−2.00 + 4.16i)10-s + (2.82 − 2.25i)11-s − 1.00i·12-s + (−2.64 − 3.31i)13-s + (−3.52 + 0.805i)14-s + (4.50 − 1.02i)15-s + (−3.02 − 3.79i)16-s − 6.61i·17-s + ⋯
L(s)  = 1  + (1.11 + 0.254i)2-s + (−0.405 − 0.841i)3-s + (0.278 + 0.134i)4-s + (−0.284 + 1.24i)5-s + (−0.237 − 1.04i)6-s + (−0.761 + 0.366i)7-s + (−0.618 − 0.492i)8-s + (0.0793 − 0.0995i)9-s + (−0.633 + 1.31i)10-s + (0.852 − 0.680i)11-s − 0.288i·12-s + (−0.732 − 0.918i)13-s + (−0.942 + 0.215i)14-s + (1.16 − 0.265i)15-s + (−0.756 − 0.948i)16-s − 1.60i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.317 + 0.948i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.317 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.781326 - 1.08541i\)
\(L(\frac12)\) \(\approx\) \(0.781326 - 1.08541i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 - 0.360i)T + (1.80 + 0.867i)T^{2} \)
3 \( 1 + (0.702 + 1.45i)T + (-1.87 + 2.34i)T^{2} \)
5 \( 1 + (0.635 - 2.78i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (2.01 - 0.970i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-2.82 + 2.25i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (2.64 + 3.31i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 6.61iT - 17T^{2} \)
19 \( 1 + (-0.804 + 1.67i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (0.720 + 3.15i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (1.06 + 0.242i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (-6.80 - 5.42i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 - 2.85iT - 41T^{2} \)
43 \( 1 + (2.69 - 0.615i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (5.47 - 4.36i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.445 + 1.94i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + (0.702 + 1.45i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (-6.52 + 8.18i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (0.952 + 1.19i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.284 + 0.0649i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + (-3.97 - 3.17i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (7.15 + 3.44i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-8.48 - 1.93i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (-7.18 + 14.9i)T + (-60.4 - 75.8i)T^{2} \)
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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

−9.865983282243302907604472220193, −9.317998346476779836653898950372, −7.80312517357243396344811633169, −6.84838735565543186282382360545, −6.53175762590985609633111809470, −5.76110836976910393864743615675, −4.63017688869680637464207446575, −3.26538715699956768298754027402, −2.84409863317501548258792133600, −0.47134792476444268192398113560, 1.83978937265980316183946182152, 3.77735304004766804104208061418, 4.10150200522458996014480493603, 4.84677681783079989248958755772, 5.67628632662782105716315843669, 6.69733002220172212439582917769, 7.961266521671800586905923798192, 9.090033971316476169399297262230, 9.573516128636878075545169310390, 10.48375713328716429810265154377

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