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Properties

Label 2-29e2-29.22-c1-0-14
Degree $2$
Conductor $841$
Sign $-0.722 - 0.691i$
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  + (0.602 − 0.137i)2-s + (−0.268 + 0.556i)3-s + (−1.45 + 0.702i)4-s + (0.857 + 3.75i)5-s + (−0.0849 + 0.372i)6-s + (2.01 + 0.970i)7-s + (−1.74 + 1.39i)8-s + (1.63 + 2.04i)9-s + (1.03 + 2.14i)10-s + (−1.08 − 0.861i)11-s − 0.999i·12-s + (0.147 − 0.184i)13-s + (1.34 + 0.307i)14-s + (−2.32 − 0.530i)15-s + (1.15 − 1.44i)16-s − 4.38i·17-s + ⋯
L(s)  = 1  + (0.426 − 0.0972i)2-s + (−0.154 + 0.321i)3-s + (−0.728 + 0.351i)4-s + (0.383 + 1.68i)5-s + (−0.0346 + 0.152i)6-s + (0.761 + 0.366i)7-s + (−0.618 + 0.492i)8-s + (0.544 + 0.682i)9-s + (0.326 + 0.678i)10-s + (−0.325 − 0.259i)11-s − 0.288i·12-s + (0.0408 − 0.0511i)13-s + (0.360 + 0.0821i)14-s + (−0.599 − 0.136i)15-s + (0.289 − 0.362i)16-s − 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.722 - 0.691i$
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: \((2,\ 841,\ (\ :1/2),\ -0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593149 + 1.47739i\)
\(L(\frac12)\) \(\approx\) \(0.593149 + 1.47739i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.602 + 0.137i)T + (1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.268 - 0.556i)T + (-1.87 - 2.34i)T^{2} \)
5 \( 1 + (-0.857 - 3.75i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (-2.01 - 0.970i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (1.08 + 0.861i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.147 + 0.184i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 4.38iT - 17T^{2} \)
19 \( 1 + (-2.10 - 4.37i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (-0.275 + 1.20i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (9.83 - 2.24i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (-3.68 + 2.93i)T + (8.23 - 36.0i)T^{2} \)
41 \( 1 + 3.85iT - 41T^{2} \)
43 \( 1 + (-7.05 - 1.61i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (-5.47 - 4.36i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (-0.445 - 1.94i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 - 6.09T + 59T^{2} \)
61 \( 1 + (0.268 - 0.556i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (-0.952 - 1.19i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (6.52 - 8.18i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (13.3 + 3.05i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (-4.76 + 3.79i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-8.95 + 4.31i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-4.59 + 1.04i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-1.54 - 3.20i)T + (-60.4 + 75.8i)T^{2} \)
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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

−10.62596789140238629011876802314, −9.816078606163824635050792755728, −8.950222457770236381750950945920, −7.72136539330966355463836871035, −7.30303339091413949275275113997, −5.84304655800974596398353553383, −5.27458069048673251551626962767, −4.15882857936807437477033626047, −3.15371891342622954475905353511, −2.18598908661274994856123770033, 0.74777230526507620978422880461, 1.66930958910473202140614148004, 3.89188825909699118911716807304, 4.57365950531642123150022631104, 5.32694917169042794239738236893, 6.06138624430064679643079823633, 7.35082858792306486311687453680, 8.308902978706365989800471135546, 9.148952049578102968618056733825, 9.581730373499390680853704278969

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