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Properties

Label 2-2301-2301.1559-c0-0-1
Degree $2$
Conductor $2301$
Sign $-0.837 + 0.545i$
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  + (−1.21 − 0.922i)2-s + (−0.468 − 0.883i)3-s + (0.354 + 1.27i)4-s + (0.156 + 0.148i)5-s + (−0.246 + 1.50i)6-s + (0.182 − 0.458i)8-s + (−0.561 + 0.827i)9-s + (−0.0533 − 0.325i)10-s + (−0.719 − 0.432i)11-s + (0.960 − 0.910i)12-s + (0.561 + 0.827i)13-s + (0.0578 − 0.208i)15-s + (0.489 − 0.294i)16-s + (1.44 − 0.486i)18-s + (−0.134 + 0.252i)20-s + ⋯
L(s)  = 1  + (−1.21 − 0.922i)2-s + (−0.468 − 0.883i)3-s + (0.354 + 1.27i)4-s + (0.156 + 0.148i)5-s + (−0.246 + 1.50i)6-s + (0.182 − 0.458i)8-s + (−0.561 + 0.827i)9-s + (−0.0533 − 0.325i)10-s + (−0.719 − 0.432i)11-s + (0.960 − 0.910i)12-s + (0.561 + 0.827i)13-s + (0.0578 − 0.208i)15-s + (0.489 − 0.294i)16-s + (1.44 − 0.486i)18-s + (−0.134 + 0.252i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2301\)    =    \(3 \cdot 13 \cdot 59\)
Sign: $-0.837 + 0.545i$
Analytic conductor: \(1.14834\)
Root analytic conductor: \(1.07161\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2301} (1559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2301,\ (\ :0),\ -0.837 + 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4440606044\)
\(L(\frac12)\) \(\approx\) \(0.4440606044\)
\(L(1)\) 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 \( 1 + (0.419 + 0.907i)T \)
good2 \( 1 + (1.21 + 0.922i)T + (0.267 + 0.963i)T^{2} \)
5 \( 1 + (-0.156 - 0.148i)T + (0.0541 + 0.998i)T^{2} \)
7 \( 1 + (0.161 + 0.986i)T^{2} \)
11 \( 1 + (0.719 + 0.432i)T + (0.468 + 0.883i)T^{2} \)
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 \( 1 + (1.14 - 0.386i)T + (0.796 - 0.605i)T^{2} \)
43 \( 1 + (-1.70 + 1.02i)T + (0.468 - 0.883i)T^{2} \)
47 \( 1 + (-1.43 + 1.35i)T + (0.0541 - 0.998i)T^{2} \)
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 \( 1 + (-1.28 + 1.21i)T + (0.0541 - 0.998i)T^{2} \)
73 \( 1 + (-0.907 + 0.419i)T^{2} \)
79 \( 1 + (0.745 - 1.40i)T + (-0.561 - 0.827i)T^{2} \)
83 \( 1 + (-1.98 + 0.215i)T + (0.976 - 0.214i)T^{2} \)
89 \( 1 + (-0.508 + 0.386i)T + (0.267 - 0.963i)T^{2} \)
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

−8.811997854468538558805148921701, −8.295557056000221252496143770339, −7.57482794553962984064031029669, −6.72648355896864017680683604376, −5.91750221724968976051219487117, −5.02702252569241150607501915860, −3.60106902751985892273572392382, −2.47259889730306654825348077453, −1.84548643053190056248417084446, −0.58261435008293396114345128595, 1.08734708251829554327520940338, 2.86929564420355004986803962674, 3.98279157183081026968394849628, 5.06552883852479066297886287263, 5.77860382333059945198308918847, 6.34749505661967322358194584058, 7.47523175839950637228008319466, 7.903813953103484696549260594014, 9.022789276063857778742024727075, 9.188436485401281519295646952510

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