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Properties

Label 2-29e2-29.23-c1-0-2
Degree $2$
Conductor $841$
Sign $0.507 + 0.861i$
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  + (−1.00 + 1.26i)2-s + (−0.360 + 1.57i)3-s + (−0.137 − 0.602i)4-s + (−1.77 + 2.23i)5-s + (−1.63 − 2.04i)6-s + (−0.497 + 2.18i)7-s + (−2.01 − 0.970i)8-s + (0.344 + 0.165i)9-s + (−1.02 − 4.50i)10-s + (3.25 − 1.56i)11-s + 1.00·12-s + (−3.81 + 1.83i)13-s + (−2.25 − 2.82i)14-s + (−2.87 − 3.61i)15-s + (4.37 − 2.10i)16-s − 6.61·17-s + ⋯
L(s)  = 1  + (−0.713 + 0.894i)2-s + (−0.207 + 0.910i)3-s + (−0.0687 − 0.301i)4-s + (−0.795 + 0.997i)5-s + (−0.666 − 0.835i)6-s + (−0.188 + 0.823i)7-s + (−0.712 − 0.343i)8-s + (0.114 + 0.0552i)9-s + (−0.324 − 1.42i)10-s + (0.982 − 0.473i)11-s + 0.288·12-s + (−1.05 + 0.509i)13-s + (−0.602 − 0.755i)14-s + (−0.743 − 0.932i)15-s + (1.09 − 0.526i)16-s − 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.507 + 0.861i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (574, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.507 + 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349820 - 0.199951i\)
\(L(\frac12)\) \(\approx\) \(0.349820 - 0.199951i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.00 - 1.26i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (0.360 - 1.57i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (1.77 - 2.23i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.497 - 2.18i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-3.25 + 1.56i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (3.81 - 1.83i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 6.61T + 17T^{2} \)
19 \( 1 + (0.412 + 1.80i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (-2.01 - 2.53i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-0.679 + 0.852i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (-7.84 - 3.77i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + 2.85T + 41T^{2} \)
43 \( 1 + (1.72 + 2.16i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-6.30 + 3.03i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.24 - 1.56i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + (0.360 - 1.57i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-9.43 - 4.54i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (1.37 - 0.662i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (0.181 + 0.228i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (4.58 + 2.20i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (1.76 + 7.74i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (5.42 - 6.80i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-3.68 - 16.1i)T + (-87.3 + 42.0i)T^{2} \)
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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

−10.86917145830930471320565764789, −9.720100805172937114742434493742, −9.202137366736663139015116191819, −8.484892878995417433215359571233, −7.29038321616619870142723510835, −6.86429787073429555139621224273, −5.96728566589635198729307518028, −4.65987311339257578284892471376, −3.72561576824890966794970101590, −2.62303060986690940438335078588, 0.29077940764599411259442238336, 1.16628943605974067972618782399, 2.32812880549225347449146164385, 3.94844209277927581173140588579, 4.73333035205305402134789563553, 6.25119139347999214366893429595, 7.07591145135674189710818788498, 7.87918247781411723107449438652, 8.798310760573978506990039158589, 9.516045837798526593672376729268

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