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Properties

Label 2-29e2-29.16-c1-0-0
Degree $2$
Conductor $841$
Sign $-0.754 + 0.655i$
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  + (−0.0377 − 0.0181i)2-s + (−1.88 + 2.36i)3-s + (−1.24 − 1.56i)4-s + (1.56 + 0.755i)5-s + (0.114 − 0.0551i)6-s + (1.28 − 1.60i)7-s + (0.0373 + 0.163i)8-s + (−1.37 − 6.01i)9-s + (−0.0455 − 0.0571i)10-s + (−0.412 + 1.80i)11-s + 6.05·12-s + (−0.573 + 2.51i)13-s + (−0.0776 + 0.0373i)14-s + (−4.75 + 2.28i)15-s + (−0.887 + 3.88i)16-s − 5.23·17-s + ⋯
L(s)  = 1  + (−0.0267 − 0.0128i)2-s + (−1.08 + 1.36i)3-s + (−0.622 − 0.781i)4-s + (0.701 + 0.338i)5-s + (0.0467 − 0.0224i)6-s + (0.483 − 0.606i)7-s + (0.0131 + 0.0578i)8-s + (−0.457 − 2.00i)9-s + (−0.0144 − 0.0180i)10-s + (−0.124 + 0.544i)11-s + 1.74·12-s + (−0.159 + 0.697i)13-s + (−0.0207 + 0.00998i)14-s + (−1.22 + 0.590i)15-s + (−0.221 + 0.972i)16-s − 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.754 + 0.655i$
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: \((2,\ 841,\ (\ :1/2),\ -0.754 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00869395 - 0.0232641i\)
\(L(\frac12)\) \(\approx\) \(0.00869395 - 0.0232641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0377 + 0.0181i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (1.88 - 2.36i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (-1.56 - 0.755i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (-1.28 + 1.60i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (0.412 - 1.80i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.573 - 2.51i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + 5.23T + 17T^{2} \)
19 \( 1 + (0.236 + 0.296i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (3.27 - 1.57i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (-2.13 - 1.02i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (1.46 + 6.40i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + 9.44T + 41T^{2} \)
43 \( 1 + (0.896 - 0.431i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-2.26 + 9.91i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (8.94 + 4.30i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + 6.24T + 59T^{2} \)
61 \( 1 + (-4.97 + 6.24i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (-0.920 - 4.03i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-0.507 + 2.22i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (8.16 - 3.93i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (-2.28 - 10.0i)T + (-71.1 + 34.2i)T^{2} \)
83 \( 1 + (9.73 + 12.2i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (8.66 + 4.17i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (3.22 + 4.04i)T + (-21.5 + 94.5i)T^{2} \)
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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

−10.56922953011231482122679483944, −9.967306457218228578320617475500, −9.480935540847519142119072891198, −8.554251419302026195412597326045, −6.90540387365713612342989595991, −6.16223395462320528056354465605, −5.26817343246253835183165737320, −4.57492154449020504283143425019, −3.95589844885221704992840131378, −1.88819078808111541818375918897, 0.01400817331457588562265934718, 1.59712585453402577872211409399, 2.77208599531635052108168242011, 4.58414670230266891216235265280, 5.41283984353546718775413041000, 6.11736478376672728709607415788, 7.04530752121112275561691979772, 8.091714777735794818123866142817, 8.486749344882059189539592899914, 9.594828843967495459205652279838

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