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Properties

Label 2-2303-2303.2020-c0-0-2
Degree $2$
Conductor $2303$
Sign $0.920 - 0.391i$
Analytic cond. $1.14934$
Root an. cond. $1.07207$
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  + (0.153 − 0.142i)2-s + (1.36 − 0.205i)3-s + (−0.0714 + 0.953i)4-s + (0.180 − 0.225i)6-s + (0.992 + 0.119i)7-s + (0.254 + 0.319i)8-s + (0.869 − 0.268i)9-s + (0.0987 + 1.31i)12-s + (0.169 − 0.122i)14-s + (−0.861 − 0.129i)16-s + (−0.270 + 0.184i)17-s + (0.0951 − 0.164i)18-s + (1.38 − 0.0413i)21-s + (0.414 + 0.384i)24-s + (−0.733 − 0.680i)25-s + ⋯
L(s)  = 1  + (0.153 − 0.142i)2-s + (1.36 − 0.205i)3-s + (−0.0714 + 0.953i)4-s + (0.180 − 0.225i)6-s + (0.992 + 0.119i)7-s + (0.254 + 0.319i)8-s + (0.869 − 0.268i)9-s + (0.0987 + 1.31i)12-s + (0.169 − 0.122i)14-s + (−0.861 − 0.129i)16-s + (−0.270 + 0.184i)17-s + (0.0951 − 0.164i)18-s + (1.38 − 0.0413i)21-s + (0.414 + 0.384i)24-s + (−0.733 − 0.680i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2303\)    =    \(7^{2} \cdot 47\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(1.14934\)
Root analytic conductor: \(1.07207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2303} (2020, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2303,\ (\ :0),\ 0.920 - 0.391i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.136147385\)
\(L(\frac12)\) \(\approx\) \(2.136147385\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.992 - 0.119i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
good2 \( 1 + (-0.153 + 0.142i)T + (0.0747 - 0.997i)T^{2} \)
3 \( 1 + (-1.36 + 0.205i)T + (0.955 - 0.294i)T^{2} \)
5 \( 1 + (0.733 + 0.680i)T^{2} \)
11 \( 1 + (-0.826 - 0.563i)T^{2} \)
13 \( 1 + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (0.270 - 0.184i)T + (0.365 - 0.930i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.365 - 0.930i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.130 + 1.74i)T + (-0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.149 - 1.99i)T + (-0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.615 + 1.56i)T + (-0.733 + 0.680i)T^{2} \)
61 \( 1 + (0.141 + 1.88i)T + (-0.988 + 0.149i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.505 + 0.243i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.0747 - 0.997i)T^{2} \)
79 \( 1 + (-0.550 - 0.954i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.0332 - 0.145i)T + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (1.54 - 0.476i)T + (0.826 - 0.563i)T^{2} \)
97 \( 1 - 1.82T + T^{2} \)
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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

−8.985150678794462859081386938423, −8.404781169179042011218243628196, −7.79101485930509841277456783948, −7.42247784737510024398282724572, −6.24229780527073836720960034036, −5.00280293769538736617525394884, −4.16449165087006333060050263489, −3.45780527447822432072784592317, −2.47601287624279780230734094506, −1.85652611225860505059497541462, 1.45564086214199453856310456730, 2.26560690020220831634506108132, 3.40769193505499241375509519723, 4.36021212849385503858638036199, 5.03438391230088947786924751851, 5.93840197261835948801078702987, 6.97788579892140032379169509752, 7.71427880203854166149233360509, 8.507562381606135858156943953803, 9.004935713356647783444923193417

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