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Properties

Label 2-2301-2301.1442-c0-0-1
Degree $2$
Conductor $2301$
Sign $0.545 - 0.837i$
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  + (−1.03 + 0.783i)2-s + (0.468 − 0.883i)3-s + (0.180 − 0.651i)4-s + (−1.44 + 1.36i)5-s + (0.209 + 1.27i)6-s + (−0.155 − 0.389i)8-s + (−0.561 − 0.827i)9-s + (0.416 − 2.54i)10-s + (1.55 − 0.935i)11-s + (−0.491 − 0.465i)12-s + (−0.561 + 0.827i)13-s + (0.531 + 1.91i)15-s + (1.04 + 0.628i)16-s + (1.22 + 0.413i)18-s + (0.629 + 1.18i)20-s + ⋯
L(s)  = 1  + (−1.03 + 0.783i)2-s + (0.468 − 0.883i)3-s + (0.180 − 0.651i)4-s + (−1.44 + 1.36i)5-s + (0.209 + 1.27i)6-s + (−0.155 − 0.389i)8-s + (−0.561 − 0.827i)9-s + (0.416 − 2.54i)10-s + (1.55 − 0.935i)11-s + (−0.491 − 0.465i)12-s + (−0.561 + 0.827i)13-s + (0.531 + 1.91i)15-s + (1.04 + 0.628i)16-s + (1.22 + 0.413i)18-s + (0.629 + 1.18i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2301\)    =    \(3 \cdot 13 \cdot 59\)
Sign: $0.545 - 0.837i$
Analytic conductor: \(1.14834\)
Root analytic conductor: \(1.07161\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2301} (1442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2301,\ (\ :0),\ 0.545 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5890696535\)
\(L(\frac12)\) \(\approx\) \(0.5890696535\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.468 + 0.883i)T \)
13 \( 1 + (0.561 - 0.827i)T \)
59 \( 1 + (0.907 + 0.419i)T \)
good2 \( 1 + (1.03 - 0.783i)T + (0.267 - 0.963i)T^{2} \)
5 \( 1 + (1.44 - 1.36i)T + (0.0541 - 0.998i)T^{2} \)
7 \( 1 + (0.161 - 0.986i)T^{2} \)
11 \( 1 + (-1.55 + 0.935i)T + (0.468 - 0.883i)T^{2} \)
17 \( 1 + (0.161 + 0.986i)T^{2} \)
19 \( 1 + (-0.647 - 0.762i)T^{2} \)
23 \( 1 + (-0.796 + 0.605i)T^{2} \)
29 \( 1 + (-0.267 - 0.963i)T^{2} \)
31 \( 1 + (-0.647 + 0.762i)T^{2} \)
37 \( 1 + (0.725 + 0.687i)T^{2} \)
41 \( 1 + (-1.50 - 0.508i)T + (0.796 + 0.605i)T^{2} \)
43 \( 1 + (-1.70 - 1.02i)T + (0.468 + 0.883i)T^{2} \)
47 \( 1 + (0.234 + 0.222i)T + (0.0541 + 0.998i)T^{2} \)
53 \( 1 + (0.947 - 0.319i)T^{2} \)
61 \( 1 + (-1.03 + 0.783i)T + (0.267 - 0.963i)T^{2} \)
67 \( 1 + (0.725 - 0.687i)T^{2} \)
71 \( 1 + (-0.680 - 0.644i)T + (0.0541 + 0.998i)T^{2} \)
73 \( 1 + (-0.907 - 0.419i)T^{2} \)
79 \( 1 + (-0.745 - 1.40i)T + (-0.561 + 0.827i)T^{2} \)
83 \( 1 + (-0.107 - 0.0117i)T + (0.976 + 0.214i)T^{2} \)
89 \( 1 + (-1.50 - 1.14i)T + (0.267 + 0.963i)T^{2} \)
97 \( 1 + (-0.907 + 0.419i)T^{2} \)
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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

−9.126631545752959289527084238973, −8.270231504175189273601557522906, −7.74494745857403001126567942252, −7.19193466937761271191323139664, −6.44373901342010888651026390232, −6.26986336308869133961286032668, −4.12176115303632587241696328458, −3.56782015449345968480360332740, −2.60430177977478898656047919825, −0.927003825573759710725937456258, 0.77480249134957263979438344638, 2.04297451078676772567990930347, 3.35722367059602547722289633918, 4.12701871653458295567951315035, 4.76053151778987690301992085947, 5.63986822844108256380456627038, 7.37548442633978936199118439856, 7.82490759054065794114101052401, 8.692173704420382005263952456358, 9.104005554791194607286549727978

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