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Properties

Label 2-621-207.160-c0-0-2
Degree $2$
Conductor $621$
Sign $-0.342 + 0.939i$
Analytic cond. $0.309919$
Root an. cond. $0.556704$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 − 1.32i)2-s + (−0.673 − 1.16i)4-s − 0.532·8-s + (−0.766 − 1.32i)13-s + (0.266 − 0.460i)16-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 2.34·26-s + (−0.939 + 1.62i)29-s + (0.939 + 1.62i)31-s + (−0.673 − 1.16i)32-s + (0.173 + 0.300i)41-s + 1.53·46-s + (0.173 − 0.300i)47-s + (−0.5 − 0.866i)49-s + (0.766 + 1.32i)50-s + ⋯
L(s)  = 1  + (0.766 − 1.32i)2-s + (−0.673 − 1.16i)4-s − 0.532·8-s + (−0.766 − 1.32i)13-s + (0.266 − 0.460i)16-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 2.34·26-s + (−0.939 + 1.62i)29-s + (0.939 + 1.62i)31-s + (−0.673 − 1.16i)32-s + (0.173 + 0.300i)41-s + 1.53·46-s + (0.173 − 0.300i)47-s + (−0.5 − 0.866i)49-s + (0.766 + 1.32i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(0.309919\)
Root analytic conductor: \(0.556704\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 621,\ (\ :0),\ -0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.311735294\)
\(L(\frac12)\) \(\approx\) \(1.311735294\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.87T + T^{2} \)
73 \( 1 + 1.87T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−10.71142561178208337640214812177, −10.04117846817129973168929527209, −9.181323178162689451948923220953, −7.928807088822871905274227282622, −7.01027298960761330497798652845, −5.43815098485986217608396666240, −4.98507722889181806500686992138, −3.57875957100403400498810604972, −2.92249441189409084744162805506, −1.51226274172792546987159182987, 2.34958866851873785127239021427, 4.13482446410873938224942989801, 4.59348575045802931286021228395, 5.86239636672206523508140271820, 6.48249028855134816428742298815, 7.41224670211874490601920601658, 8.098693657401805342250359967628, 9.190758642907637776843241152064, 10.04820021977491107690521593635, 11.28735616674712518041618037531

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