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Properties

Label 2-2301-2301.1598-c0-0-1
Degree $2$
Conductor $2301$
Sign $-0.550 + 0.834i$
Analytic cond. $1.14834$
Root an. cond. $1.07161$
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  + (−1.72 − 0.579i)2-s + (−0.856 − 0.515i)3-s + (1.82 + 1.38i)4-s + (−0.0400 + 0.100i)5-s + (1.17 + 1.38i)6-s + (−1.31 − 1.94i)8-s + (0.468 + 0.883i)9-s + (0.127 − 0.149i)10-s + (0.522 − 1.88i)11-s + (−0.849 − 2.13i)12-s + (0.468 − 0.883i)13-s + (0.0861 − 0.0655i)15-s + (0.527 + 1.89i)16-s + (−0.293 − 1.79i)18-s + (−0.212 + 0.128i)20-s + ⋯
L(s)  = 1  + (−1.72 − 0.579i)2-s + (−0.856 − 0.515i)3-s + (1.82 + 1.38i)4-s + (−0.0400 + 0.100i)5-s + (1.17 + 1.38i)6-s + (−1.31 − 1.94i)8-s + (0.468 + 0.883i)9-s + (0.127 − 0.149i)10-s + (0.522 − 1.88i)11-s + (−0.849 − 2.13i)12-s + (0.468 − 0.883i)13-s + (0.0861 − 0.0655i)15-s + (0.527 + 1.89i)16-s + (−0.293 − 1.79i)18-s + (−0.212 + 0.128i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2301\)    =    \(3 \cdot 13 \cdot 59\)
Sign: $-0.550 + 0.834i$
Analytic conductor: \(1.14834\)
Root analytic conductor: \(1.07161\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2301} (1598, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2301,\ (\ :0),\ -0.550 + 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3834959356\)
\(L(\frac12)\) \(\approx\) \(0.3834959356\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.856 + 0.515i)T \)
13 \( 1 + (-0.468 + 0.883i)T \)
59 \( 1 + (-0.976 + 0.214i)T \)
good2 \( 1 + (1.72 + 0.579i)T + (0.796 + 0.605i)T^{2} \)
5 \( 1 + (0.0400 - 0.100i)T + (-0.725 - 0.687i)T^{2} \)
7 \( 1 + (-0.647 + 0.762i)T^{2} \)
11 \( 1 + (-0.522 + 1.88i)T + (-0.856 - 0.515i)T^{2} \)
17 \( 1 + (-0.647 - 0.762i)T^{2} \)
19 \( 1 + (-0.907 + 0.419i)T^{2} \)
23 \( 1 + (0.947 + 0.319i)T^{2} \)
29 \( 1 + (-0.796 + 0.605i)T^{2} \)
31 \( 1 + (-0.907 - 0.419i)T^{2} \)
37 \( 1 + (0.370 + 0.928i)T^{2} \)
41 \( 1 + (-0.306 - 1.87i)T + (-0.947 + 0.319i)T^{2} \)
43 \( 1 + (-0.0289 - 0.104i)T + (-0.856 + 0.515i)T^{2} \)
47 \( 1 + (0.479 + 1.20i)T + (-0.725 + 0.687i)T^{2} \)
53 \( 1 + (0.161 - 0.986i)T^{2} \)
61 \( 1 + (1.72 + 0.579i)T + (0.796 + 0.605i)T^{2} \)
67 \( 1 + (0.370 - 0.928i)T^{2} \)
71 \( 1 + (-0.634 - 1.59i)T + (-0.725 + 0.687i)T^{2} \)
73 \( 1 + (-0.976 + 0.214i)T^{2} \)
79 \( 1 + (-1.62 + 0.977i)T + (0.468 - 0.883i)T^{2} \)
83 \( 1 + (0.0786 + 1.44i)T + (-0.994 + 0.108i)T^{2} \)
89 \( 1 + (-0.306 + 0.103i)T + (0.796 - 0.605i)T^{2} \)
97 \( 1 + (-0.976 - 0.214i)T^{2} \)
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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

−8.833216781726130051380070763453, −8.270348170699172702506460769533, −7.68381175091655428485060201481, −6.72131498932428684931825290754, −6.16238686830895514347721716529, −5.24452649249442637847349987051, −3.59514433413934591060013079044, −2.80235729716925633109751574625, −1.44643612911274066857027333893, −0.63949916273934517686936750831, 1.18434454579878498279780502165, 2.18614480427142443480734293015, 4.03722321808613065745229097799, 4.81302613569371903856981870039, 5.89527667767758842124657049848, 6.69599626840512595553124177940, 7.05377899607669786222182424321, 7.916869541491792727635866991916, 9.102727280630925133625222500742, 9.236789729844885935393327413139

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