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Properties

Label 2-29e2-29.28-c1-0-0
Degree $2$
Conductor $841$
Sign $0.0204 - 0.999i$
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  − 1.80i·2-s + 0.445i·3-s − 1.24·4-s − 0.356·5-s + 0.801·6-s − 4.04·7-s − 1.35i·8-s + 2.80·9-s + 0.643i·10-s + 2.91i·11-s − 0.554i·12-s − 5.18·13-s + 7.29i·14-s − 0.158i·15-s − 4.93·16-s + 1.10i·17-s + ⋯
L(s)  = 1  − 1.27i·2-s + 0.256i·3-s − 0.623·4-s − 0.159·5-s + 0.327·6-s − 1.53·7-s − 0.479i·8-s + 0.933·9-s + 0.203i·10-s + 0.877i·11-s − 0.160i·12-s − 1.43·13-s + 1.94i·14-s − 0.0410i·15-s − 1.23·16-s + 0.269i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0204 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0204 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.0204 - 0.999i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (840, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.0204 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0975411 + 0.0955701i\)
\(L(\frac12)\) \(\approx\) \(0.0975411 + 0.0955701i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.80iT - 2T^{2} \)
3 \( 1 - 0.445iT - 3T^{2} \)
5 \( 1 + 0.356T + 5T^{2} \)
7 \( 1 + 4.04T + 7T^{2} \)
11 \( 1 - 2.91iT - 11T^{2} \)
13 \( 1 + 5.18T + 13T^{2} \)
17 \( 1 - 1.10iT - 17T^{2} \)
19 \( 1 - 2.04iT - 19T^{2} \)
23 \( 1 + 4.13T + 23T^{2} \)
31 \( 1 + 6.35iT - 31T^{2} \)
37 \( 1 - 2.91iT - 37T^{2} \)
41 \( 1 + 0.396iT - 41T^{2} \)
43 \( 1 - 5.74iT - 43T^{2} \)
47 \( 1 - 7.80iT - 47T^{2} \)
53 \( 1 + 4.35T + 53T^{2} \)
59 \( 1 + 9.10T + 59T^{2} \)
61 \( 1 + 6.04iT - 61T^{2} \)
67 \( 1 + 0.374T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 8.94iT - 73T^{2} \)
79 \( 1 + 0.594iT - 79T^{2} \)
83 \( 1 + 9.43T + 83T^{2} \)
89 \( 1 - 1.42iT - 89T^{2} \)
97 \( 1 + 15.7iT - 97T^{2} \)
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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

−10.10760452538148679383262142486, −9.729238923558019147849931503630, −9.544188402666455254823370631103, −7.78825261433071484803056836498, −7.00518462253414924748680704298, −6.11538823635522730422217889032, −4.55433679383011011343857713984, −3.90292686737028882998879577419, −2.86129923982311047396299629010, −1.83082706879780840088931908605, 0.06112259549857286529746447243, 2.38984809545065968487931630585, 3.66015231362953174897310148166, 4.90607357316287714778865667313, 5.89276339455111756813736157083, 6.68277787640735105013479416858, 7.21822887216845549545999683890, 7.974739967848191015026165590016, 9.071018855242526913552949589963, 9.762381898679518163184690721514

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