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Properties

Label 2-29e2-29.23-c1-0-52
Degree $2$
Conductor $841$
Sign $-0.990 - 0.140i$
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  + (1.39 − 1.74i)2-s + (0.497 − 2.18i)3-s + (−0.667 − 2.92i)4-s + (1.87 − 2.34i)5-s + (−3.11 − 3.90i)6-s + (−0.445 + 1.94i)7-s + (−2.01 − 0.970i)8-s + (−1.80 − 0.867i)9-s + (−1.49 − 6.54i)10-s + (−2.01 + 0.970i)11-s − 6.70·12-s + (−0.900 + 0.433i)13-s + (2.78 + 3.49i)14-s + (−4.18 − 5.24i)15-s + (0.900 − 0.433i)16-s − 4.47·17-s + ⋯
L(s)  = 1  + (0.985 − 1.23i)2-s + (0.287 − 1.25i)3-s + (−0.333 − 1.46i)4-s + (0.836 − 1.04i)5-s + (−1.27 − 1.59i)6-s + (−0.168 + 0.736i)7-s + (−0.712 − 0.343i)8-s + (−0.600 − 0.289i)9-s + (−0.472 − 2.06i)10-s + (−0.607 + 0.292i)11-s − 1.93·12-s + (−0.249 + 0.120i)13-s + (0.745 + 0.934i)14-s + (−1.07 − 1.35i)15-s + (0.225 − 0.108i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.990 - 0.140i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (574, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.990 - 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.217813 + 3.08604i\)
\(L(\frac12)\) \(\approx\) \(0.217813 + 3.08604i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.39 + 1.74i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (-0.497 + 2.18i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (-1.87 + 2.34i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.445 - 1.94i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (2.01 - 0.970i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.900 - 0.433i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (-3.74 - 4.69i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-4.18 + 5.24i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (23.0 + 28.9i)T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + (4.18 + 5.24i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-2.01 + 0.970i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (5.61 - 7.03i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + (2.98 - 13.0i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-7.20 - 3.47i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (-6.04 - 2.91i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (-1.33 - 5.84i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (2.78 - 3.49i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-2.98 - 13.0i)T + (-87.3 + 42.0i)T^{2} \)
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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

−9.781494657105713682402110768175, −9.103312112820898745756184407240, −8.158504168656105540356673689337, −7.07724935124275885047681115204, −5.89991974151880097053326797168, −5.23624588728344541907188222638, −4.30632011100804885254128576362, −2.67489130042778433254964426720, −2.15056065737779308375714417858, −1.16135433419596936128175994157, 2.71251453505139477973710234628, 3.58444043699501586942136957822, 4.59370986217862753809345801749, 5.19300314476474058507899848716, 6.46391139176848703080161017516, 6.75128869442992238186869960742, 7.88244545478262704341479807169, 8.902596188547173531608782129814, 9.954595900517937991525597407173, 10.49880623216200988835640070992

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