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Properties

Label 2-29e2-29.20-c1-0-31
Degree $2$
Conductor $841$
Sign $0.897 + 0.440i$
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  + (−0.373 + 0.179i)2-s + (−0.258 − 0.323i)3-s + (−1.14 + 1.42i)4-s + (0.900 − 0.433i)5-s + (0.154 + 0.0744i)6-s + (1.76 + 2.21i)7-s + (0.352 − 1.54i)8-s + (0.629 − 2.75i)9-s + (−0.258 + 0.323i)10-s + (−0.537 − 2.35i)11-s + 0.757·12-s + (−0.406 − 1.78i)13-s + (−1.05 − 0.508i)14-s + (−0.373 − 0.179i)15-s + (−0.667 − 2.92i)16-s − 4.82·17-s + ⋯
L(s)  = 1  + (−0.263 + 0.127i)2-s + (−0.149 − 0.186i)3-s + (−0.570 + 0.714i)4-s + (0.402 − 0.194i)5-s + (0.0631 + 0.0303i)6-s + (0.666 + 0.835i)7-s + (0.124 − 0.546i)8-s + (0.209 − 0.919i)9-s + (−0.0816 + 0.102i)10-s + (−0.161 − 0.709i)11-s + 0.218·12-s + (−0.112 − 0.494i)13-s + (−0.282 − 0.135i)14-s + (−0.0963 − 0.0464i)15-s + (−0.166 − 0.731i)16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.897 + 0.440i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.897 + 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19128 - 0.276771i\)
\(L(\frac12)\) \(\approx\) \(1.19128 - 0.276771i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.373 - 0.179i)T + (1.24 - 1.56i)T^{2} \)
3 \( 1 + (0.258 + 0.323i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-0.900 + 0.433i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (-1.76 - 2.21i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (0.537 + 2.35i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.406 + 1.78i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 + (-3.74 + 4.69i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-6.89 - 3.32i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (-3.66 + 1.76i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (-0.890 + 3.89i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 - 12.4T + 41T^{2} \)
43 \( 1 + (5.77 + 2.78i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.16 + 5.11i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-6.74 + 3.24i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 + (-0.516 - 0.647i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-1.25 + 5.51i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-0.705 - 3.09i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (3.60 + 1.73i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (0.0921 - 0.403i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (2.28 - 2.85i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (4.04 - 1.94i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (7.78 - 9.76i)T + (-21.5 - 94.5i)T^{2} \)
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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

−9.818932961463903689153006710391, −9.014505499256111663608859110819, −8.748850805193870397906731697995, −7.62241793973238226860682984167, −6.83628538504270198013161070106, −5.63948017770207630351029436899, −4.93010016344318944809037237290, −3.66901919993013094405462246209, −2.56996292500046844146149735898, −0.792948669759499669336840440010, 1.28218722185610260080672308533, 2.36580081380291123328599448126, 4.40210407870455286831331859621, 4.66192695467383077881124275538, 5.75768296274735442371573316212, 6.87581863788162857649625049384, 7.77065377149418489698354946394, 8.655917747233898463925237371638, 9.688660482120982748880229556752, 10.21888973929337115282511083160

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