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Properties

Label 2-29e2-29.24-c1-0-50
Degree $2$
Conductor $841$
Sign $-0.357 - 0.934i$
Analytic cond. $6.71541$
Root an. cond. $2.59141$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.777 − 0.974i)2-s + (0.400 + 1.75i)3-s + (0.0990 − 0.433i)4-s + (−2.52 − 3.16i)5-s + (1.40 − 1.75i)6-s + (−0.153 − 0.674i)7-s + (−2.74 + 1.32i)8-s + (−0.222 + 0.107i)9-s + (−1.12 + 4.92i)10-s + (−2.56 − 1.23i)11-s + 0.801·12-s + (−1.32 − 0.636i)13-s + (−0.538 + 0.674i)14-s + (4.54 − 5.70i)15-s + (2.62 + 1.26i)16-s + 1.60·17-s + ⋯
L(s)  = 1  + (−0.549 − 0.689i)2-s + (0.231 + 1.01i)3-s + (0.0495 − 0.216i)4-s + (−1.12 − 1.41i)5-s + (0.571 − 0.717i)6-s + (−0.0582 − 0.255i)7-s + (−0.971 + 0.467i)8-s + (−0.0741 + 0.0357i)9-s + (−0.355 + 1.55i)10-s + (−0.774 − 0.372i)11-s + 0.231·12-s + (−0.366 − 0.176i)13-s + (−0.143 + 0.180i)14-s + (1.17 − 1.47i)15-s + (0.655 + 0.315i)16-s + 0.388·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.357 - 0.934i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (778, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.357 - 0.934i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.777 + 0.974i)T + (-0.445 + 1.94i)T^{2} \)
3 \( 1 + (-0.400 - 1.75i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (2.52 + 3.16i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (0.153 + 0.674i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (2.56 + 1.23i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (1.32 + 0.636i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 - 1.60T + 17T^{2} \)
19 \( 1 + (0.599 - 2.62i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (3.21 - 4.03i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (-1.21 - 1.52i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (-2.56 + 1.23i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + 6.49T + 41T^{2} \)
43 \( 1 + (0.414 - 0.519i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (4.28 + 2.06i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-0.0304 - 0.0382i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + 6.39T + 59T^{2} \)
61 \( 1 + (0.291 + 1.27i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (13.4 - 6.48i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-2.03 - 0.980i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (5.18 - 6.50i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (8.77 - 4.22i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (-0.894 + 3.92i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (7.01 + 8.80i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (1.01 - 4.45i)T + (-87.3 - 42.0i)T^{2} \)
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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

−9.809185730515073976741146716367, −8.863814821827089577577708834481, −8.324701522967737767923109669027, −7.43749074762752207151109012123, −5.71292285206123937304476641515, −4.96337201078001514148357893639, −4.03879796757274857374233205486, −3.15379910378573583748937033629, −1.37658448710882731166610902352, 0, 2.41326671524139963569764086469, 3.12602475867105885058972369658, 4.43455303206271487347913428400, 6.18837324571673176639683754124, 6.80823188658055002544861885619, 7.53294286123461978819784702273, 7.81981382039246439125262142470, 8.645694640920682654856041026281, 9.930235025601991614168385621326

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