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Properties

Label 2-399-133.100-c1-0-11
Degree $2$
Conductor $399$
Sign $-0.448 - 0.893i$
Analytic cond. $3.18603$
Root an. cond. $1.78494$
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  + (−0.486 + 2.75i)2-s + (−0.939 + 0.342i)3-s + (−5.48 − 1.99i)4-s + (3.19 − 1.16i)5-s + (−0.486 − 2.75i)6-s + (2.40 + 1.10i)7-s + (5.36 − 9.29i)8-s + (0.766 − 0.642i)9-s + (1.65 + 9.36i)10-s + 3.52·11-s + 5.83·12-s + (−0.169 − 0.958i)13-s + (−4.22 + 6.08i)14-s + (−2.60 + 2.18i)15-s + (14.0 + 11.8i)16-s + (−1.78 − 1.49i)17-s + ⋯
L(s)  = 1  + (−0.343 + 1.94i)2-s + (−0.542 + 0.197i)3-s + (−2.74 − 0.997i)4-s + (1.42 − 0.519i)5-s + (−0.198 − 1.12i)6-s + (0.907 + 0.419i)7-s + (1.89 − 3.28i)8-s + (0.255 − 0.214i)9-s + (0.522 + 2.96i)10-s + 1.06·11-s + 1.68·12-s + (−0.0468 − 0.265i)13-s + (−1.12 + 1.62i)14-s + (−0.671 + 0.563i)15-s + (3.51 + 2.95i)16-s + (−0.432 − 0.362i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $-0.448 - 0.893i$
Analytic conductor: \(3.18603\)
Root analytic conductor: \(1.78494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :1/2),\ -0.448 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.621575 + 1.00686i\)
\(L(\frac12)\) \(\approx\) \(0.621575 + 1.00686i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-2.40 - 1.10i)T \)
19 \( 1 + (-2.46 + 3.59i)T \)
good2 \( 1 + (0.486 - 2.75i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (-3.19 + 1.16i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 - 3.52T + 11T^{2} \)
13 \( 1 + (0.169 + 0.958i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.78 + 1.49i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.270 - 1.53i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (4.62 + 1.68i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-1.30 + 2.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.89 - 5.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.257 - 1.46i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-5.57 - 4.67i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-0.474 + 0.398i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (5.94 + 2.16i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (4.71 + 3.95i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.26 - 7.20i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.45 - 8.24i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (7.41 + 6.22i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (8.47 - 3.08i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-0.591 - 0.496i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-5.63 - 9.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.267 - 0.0975i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-13.0 + 4.75i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.49707159890267429619847065662, −10.14817424508759446556138587819, −9.270765757598444196581251107147, −8.969388475415667352354143263222, −7.74893695448868579864928785990, −6.66888190846794909400289735870, −5.87995529153524599412298423083, −5.20780557834349305328383811293, −4.44733663104488297599770435854, −1.28639912316696667642038561257, 1.39361706820134421821816392814, 2.08804995012190263649561975641, 3.67089694703756363670970996166, 4.79992150997686962525328394307, 5.91603385697408031492104664254, 7.43023435334438116289854995892, 8.784557417924799852396532487427, 9.490845476784542633785335102020, 10.41588444964097932465413855531, 10.85008632959846368266246814640

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