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Properties

Label 2-29e2-29.16-c1-0-12
Degree $2$
Conductor $841$
Sign $0.592 - 0.805i$
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  + (0.460 + 0.221i)2-s + (−1.67 + 2.10i)3-s + (−1.08 − 1.35i)4-s + (−1.98 − 0.956i)5-s + (−1.24 + 0.597i)6-s + (−0.811 + 1.01i)7-s + (−0.425 − 1.86i)8-s + (−0.945 − 4.14i)9-s + (−0.702 − 0.880i)10-s + (1.11 − 4.88i)11-s + 4.68·12-s + (−0.698 + 3.05i)13-s + (−0.599 + 0.288i)14-s + (5.34 − 2.57i)15-s + (−0.556 + 2.43i)16-s + 1.37·17-s + ⋯
L(s)  = 1  + (0.325 + 0.156i)2-s + (−0.969 + 1.21i)3-s + (−0.542 − 0.679i)4-s + (−0.887 − 0.427i)5-s + (−0.506 + 0.243i)6-s + (−0.306 + 0.384i)7-s + (−0.150 − 0.658i)8-s + (−0.315 − 1.38i)9-s + (−0.222 − 0.278i)10-s + (0.336 − 1.47i)11-s + 1.35·12-s + (−0.193 + 0.848i)13-s + (−0.160 + 0.0771i)14-s + (1.38 − 0.664i)15-s + (−0.139 + 0.609i)16-s + 0.333·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.592 - 0.805i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.592 - 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.682635 + 0.345495i\)
\(L(\frac12)\) \(\approx\) \(0.682635 + 0.345495i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.460 - 0.221i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (1.67 - 2.10i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (1.98 + 0.956i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (0.811 - 1.01i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (-1.11 + 4.88i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.698 - 3.05i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
19 \( 1 + (-3.71 - 4.65i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (0.779 - 0.375i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (-2.47 - 1.19i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (-1.31 - 5.75i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 - 1.90T + 41T^{2} \)
43 \( 1 + (-9.92 + 4.77i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (0.640 - 2.80i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-5.22 - 2.51i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + 4.35T + 59T^{2} \)
61 \( 1 + (-3.38 + 4.24i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (-0.525 - 2.30i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-3.46 + 15.1i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.06 + 1.47i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (-0.470 - 2.06i)T + (-71.1 + 34.2i)T^{2} \)
83 \( 1 + (-3.96 - 4.97i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-10.7 - 5.17i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-7.12 - 8.93i)T + (-21.5 + 94.5i)T^{2} \)
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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

−10.35373712985647520072810050862, −9.498802750021082014315338464862, −8.963115088884945133160696245922, −7.901208792146600152503130890078, −6.31867878273436452811723633610, −5.80977829435816542949912769772, −4.98021278624795555699663033931, −4.13296477419803035682870362290, −3.50016853408615116700842645200, −0.802838556804899515650354605807, 0.61979902047726367547461172862, 2.47189452496659429413503512784, 3.69873846896565058288850219286, 4.69172958065552248888095872235, 5.65203617303139263285849468599, 6.92687215646977336730254245829, 7.41074581748794955722318944201, 7.87505922058824426537204261374, 9.226543505660923711271079047125, 10.24318798197215151161951225249

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