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Properties

Label 2-29e2-29.22-c1-0-39
Degree $2$
Conductor $841$
Sign $-0.834 + 0.550i$
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  + (−1.21 + 0.277i)2-s + (−0.781 + 1.62i)3-s + (−0.400 + 0.193i)4-s + (−0.900 − 3.94i)5-s + (0.5 − 2.19i)6-s + (−0.623 − 0.300i)7-s + (2.38 − 1.90i)8-s + (−0.153 − 0.193i)9-s + (2.19 + 4.54i)10-s + (2.22 + 1.77i)11-s − 0.801i·12-s + (−0.914 + 1.14i)13-s + (0.841 + 0.192i)14-s + (7.11 + 1.62i)15-s + (−1.81 + 2.27i)16-s + 1.60i·17-s + ⋯
L(s)  = 1  + (−0.859 + 0.196i)2-s + (−0.451 + 0.937i)3-s + (−0.200 + 0.0965i)4-s + (−0.402 − 1.76i)5-s + (0.204 − 0.894i)6-s + (−0.235 − 0.113i)7-s + (0.842 − 0.672i)8-s + (−0.0513 − 0.0643i)9-s + (0.692 + 1.43i)10-s + (0.672 + 0.535i)11-s − 0.231i·12-s + (−0.253 + 0.318i)13-s + (0.224 + 0.0513i)14-s + (1.83 + 0.419i)15-s + (−0.453 + 0.569i)16-s + 0.388i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.834 + 0.550i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.834 + 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0190543 - 0.0635241i\)
\(L(\frac12)\) \(\approx\) \(0.0190543 - 0.0635241i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 - 0.277i)T + (1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.781 - 1.62i)T + (-1.87 - 2.34i)T^{2} \)
5 \( 1 + (0.900 + 3.94i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (0.623 + 0.300i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-2.22 - 1.77i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.914 - 1.14i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 - 1.60iT - 17T^{2} \)
19 \( 1 + (1.16 + 2.42i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (-1.14 + 5.02i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-1.90 + 0.434i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (2.22 - 1.77i)T + (8.23 - 36.0i)T^{2} \)
41 \( 1 - 6.49iT - 41T^{2} \)
43 \( 1 + (-0.648 - 0.147i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (3.71 + 2.96i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.0108 + 0.0476i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + 6.39T + 59T^{2} \)
61 \( 1 + (-0.567 + 1.17i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (9.32 + 11.6i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (-1.40 + 1.76i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (8.11 + 1.85i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (7.61 - 6.07i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-3.62 + 1.74i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (10.9 - 2.50i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-1.98 - 4.11i)T + (-60.4 + 75.8i)T^{2} \)
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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

−9.663219267545759104185788610163, −9.074091616750246637690461102101, −8.456061278812134510539018628237, −7.60285886426042845815156689150, −6.45951314866728218076064907941, −4.95519194508969131539119784872, −4.59621054347539617754498737218, −3.87594311750563917333541345747, −1.46439558933759293820720856927, −0.05120959140112220478240916164, 1.50853177409946185170160804494, 2.87223386731660434886464384798, 3.98210562108608853298162399556, 5.66600884322610637397491847694, 6.44599853522766978088135868388, 7.29418736249500772717669000521, 7.74185922767140551608038146230, 8.906199503361348497846249743917, 9.848984007047217561674644535915, 10.49236098675638622048170938528

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