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Properties

Label 2-29e2-29.24-c1-0-46
Degree $2$
Conductor $841$
Sign $-0.463 + 0.886i$
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  + (0.258 + 0.323i)2-s + (0.0921 + 0.403i)3-s + (0.406 − 1.78i)4-s + (−0.623 − 0.781i)5-s + (−0.106 + 0.134i)6-s + (−0.629 − 2.75i)7-s + (1.42 − 0.688i)8-s + (2.54 − 1.22i)9-s + (0.0921 − 0.403i)10-s + (−2.17 − 1.04i)11-s + 0.757·12-s + (−1.64 − 0.793i)13-s + (0.730 − 0.915i)14-s + (0.258 − 0.323i)15-s + (−2.70 − 1.30i)16-s − 4.82·17-s + ⋯
L(s)  = 1  + (0.182 + 0.228i)2-s + (0.0532 + 0.233i)3-s + (0.203 − 0.891i)4-s + (−0.278 − 0.349i)5-s + (−0.0436 + 0.0547i)6-s + (−0.237 − 1.04i)7-s + (0.505 − 0.243i)8-s + (0.849 − 0.409i)9-s + (0.0291 − 0.127i)10-s + (−0.655 − 0.315i)11-s + 0.218·12-s + (−0.456 − 0.220i)13-s + (0.195 − 0.244i)14-s + (0.0666 − 0.0836i)15-s + (−0.675 − 0.325i)16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.463 + 0.886i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (778, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.463 + 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.668773 - 1.10476i\)
\(L(\frac12)\) \(\approx\) \(0.668773 - 1.10476i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.323i)T + (-0.445 + 1.94i)T^{2} \)
3 \( 1 + (-0.0921 - 0.403i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (0.623 + 0.781i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (0.629 + 2.75i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (2.17 + 1.04i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (1.64 + 0.793i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 + (1.33 - 5.84i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (4.77 - 5.98i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (2.53 + 3.18i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (-3.60 + 1.73i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 - 12.4T + 41T^{2} \)
43 \( 1 + (-3.99 + 5.01i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (4.72 + 2.27i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (4.66 + 5.85i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 + (0.184 + 0.807i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-5.09 + 2.45i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-2.85 - 1.37i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.49 + 3.12i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (0.373 - 0.179i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (-0.813 + 3.56i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-2.79 - 3.50i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (-2.77 + 12.1i)T + (-87.3 - 42.0i)T^{2} \)
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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

−9.998694562193329822601355701690, −9.365506961645962892988998575392, −8.014806468485945375017177632711, −7.32703037558414671185985709981, −6.41321820917091084855854882670, −5.53624574396182196575491549159, −4.39341725801896737540788581859, −3.85430235948143297168556762503, −2.02103877084818031845182020650, −0.56054857004110585455592721097, 2.29504996597483172484529119019, 2.65832785419631649050235224406, 4.20453603066542444413563143946, 4.87021469257143927532298640558, 6.36358206360934907477442241206, 7.12484309363366853813432500571, 7.83287854849112050966938058760, 8.734699900462957995086057295624, 9.540015087932481190738172242427, 10.77445691346822955391771208921

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