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Properties

Label 2-29e2-29.5-c1-0-21
Degree $2$
Conductor $841$
Sign $0.959 + 0.280i$
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  + (−0.483 + 0.385i)2-s + (−0.602 + 0.137i)3-s + (−0.360 + 1.57i)4-s + (−2.40 − 3.01i)5-s + (0.238 − 0.298i)6-s + (0.497 + 2.18i)7-s + (−0.970 − 2.01i)8-s + (−2.35 + 1.13i)9-s + (2.32 + 0.530i)10-s + (−0.599 + 1.24i)11-s − i·12-s + (−0.212 − 0.102i)13-s + (−1.08 − 0.861i)14-s + (1.86 + 1.48i)15-s + (−1.67 − 0.804i)16-s − 4.38i·17-s + ⋯
L(s)  = 1  + (−0.341 + 0.272i)2-s + (−0.347 + 0.0794i)3-s + (−0.180 + 0.788i)4-s + (−1.07 − 1.34i)5-s + (0.0972 − 0.121i)6-s + (0.188 + 0.823i)7-s + (−0.343 − 0.712i)8-s + (−0.786 + 0.378i)9-s + (0.734 + 0.167i)10-s + (−0.180 + 0.375i)11-s − 0.288i·12-s + (−0.0589 − 0.0284i)13-s + (−0.288 − 0.230i)14-s + (0.480 + 0.383i)15-s + (−0.417 − 0.201i)16-s − 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.959 + 0.280i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.959 + 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.622336 - 0.0891486i\)
\(L(\frac12)\) \(\approx\) \(0.622336 - 0.0891486i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.483 - 0.385i)T + (0.445 - 1.94i)T^{2} \)
3 \( 1 + (0.602 - 0.137i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (2.40 + 3.01i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (-0.497 - 2.18i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (0.599 - 1.24i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.212 + 0.102i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + 4.38iT - 17T^{2} \)
19 \( 1 + (-4.73 - 1.08i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (0.770 - 0.966i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (-7.88 + 6.29i)T + (6.89 - 30.2i)T^{2} \)
37 \( 1 + (-2.04 - 4.24i)T + (-23.0 + 28.9i)T^{2} \)
41 \( 1 + 3.85iT - 41T^{2} \)
43 \( 1 + (5.65 + 4.51i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-3.03 + 6.30i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.24 + 1.56i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 - 6.09T + 59T^{2} \)
61 \( 1 + (0.602 - 0.137i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (1.37 - 0.662i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-9.43 - 4.54i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-10.7 - 8.54i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (-2.64 - 5.48i)T + (-49.2 + 61.7i)T^{2} \)
83 \( 1 + (-2.21 + 9.69i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (3.68 - 2.93i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + (-3.47 - 0.792i)T + (87.3 + 42.0i)T^{2} \)
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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

−9.832348261910190735039300449071, −9.100904833836167295903142168594, −8.275370589070684939729164820063, −7.990778815317527424748306812936, −6.97684769378808574787813411443, −5.48313722327801329065921412930, −4.92794029567890331250617640142, −3.90113115861377640154787546920, −2.65719264078806938186013227484, −0.51566302638163570410710263818, 0.889337828841345916381451073173, 2.75928883465164406320496346467, 3.68077664947873830120129260825, 4.87403521096344484593562204629, 6.10111557418630117440157518870, 6.68869530819481486162210016170, 7.75511910666771003929141104723, 8.443171301947526139891541334559, 9.631543538772115195636821705381, 10.57377543277009793512584360486

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