Properties

Label 2-29e2-29.4-c1-0-37
Degree $2$
Conductor $841$
Sign $0.864 + 0.501i$
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  + (2.35 + 0.537i)2-s + (−1.04 − 2.17i)3-s + (3.44 + 1.66i)4-s + (−0.222 + 0.974i)5-s + (−1.29 − 5.68i)6-s + (2.54 − 1.22i)7-s + (3.45 + 2.75i)8-s + (−1.76 + 2.21i)9-s + (−1.04 + 2.17i)10-s + (0.323 − 0.258i)11-s − 9.24i·12-s + (2.38 + 2.99i)13-s + (6.65 − 1.51i)14-s + (2.35 − 0.537i)15-s + (1.87 + 2.34i)16-s + 0.828i·17-s + ⋯
L(s)  = 1  + (1.66 + 0.379i)2-s + (−0.604 − 1.25i)3-s + (1.72 + 0.830i)4-s + (−0.0995 + 0.436i)5-s + (−0.529 − 2.31i)6-s + (0.963 − 0.463i)7-s + (1.22 + 0.973i)8-s + (−0.587 + 0.737i)9-s + (−0.331 + 0.687i)10-s + (0.0976 − 0.0778i)11-s − 2.66i·12-s + (0.662 + 0.830i)13-s + (1.77 − 0.406i)14-s + (0.607 − 0.138i)15-s + (0.467 + 0.586i)16-s + 0.200i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.864 + 0.501i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.864 + 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.51034 - 0.944457i\)
\(L(\frac12)\) \(\approx\) \(3.51034 - 0.944457i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.35 - 0.537i)T + (1.80 + 0.867i)T^{2} \)
3 \( 1 + (1.04 + 2.17i)T + (-1.87 + 2.34i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (-2.54 + 1.22i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-0.323 + 0.258i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (-2.38 - 2.99i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 0.828iT - 17T^{2} \)
19 \( 1 + (-2.60 + 5.40i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (0.813 + 3.56i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (9.81 + 2.24i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (-3.12 - 2.49i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 - 4.48iT - 41T^{2} \)
43 \( 1 + (-3.49 + 0.797i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (2.53 - 2.02i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (2.11 - 9.24i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 3.65T + 59T^{2} \)
61 \( 1 + (-2.09 - 4.35i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (3.52 - 4.42i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-5.50 - 6.90i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (3.89 - 0.890i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + (1.88 + 1.50i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (6.89 + 3.32i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-12.1 - 2.77i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (1.94 - 4.04i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.88295190606124132385092666760, −9.089645111975705744755481299800, −7.78828854484648530138832501332, −7.17288753342412048660283866327, −6.58316981266034410041590783412, −5.82356020384711580818836880006, −4.84570300945459100825483647574, −3.99326496267774278669912434047, −2.67106685423758962235281285310, −1.38803769438630638393140045290, 1.74545550923616741903269325642, 3.36487921705165706217179104460, 4.00084343990645630425813355064, 5.06684916322179753409228883913, 5.32774917490225906050309145938, 6.09539225004731141024726138157, 7.60444789010429154126119615710, 8.698013460386253706084512128902, 9.753016925059137579642303658262, 10.75799980384828327414775413477

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