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Properties

Label 2-2385-265.52-c0-0-3
Degree $2$
Conductor $2385$
Sign $0.997 + 0.0746i$
Analytic cond. $1.19027$
Root an. cond. $1.09099$
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  + (0.437 − 0.437i)2-s + 0.618i·4-s + (−0.987 + 0.156i)5-s + (−0.642 − 0.642i)7-s + (0.707 + 0.707i)8-s + (−0.363 + 0.5i)10-s + (1.39 − 1.39i)13-s − 0.561·14-s + (−0.0966 − 0.610i)20-s + (1.34 + 1.34i)23-s + (0.951 − 0.309i)25-s − 1.22i·26-s + (0.396 − 0.396i)28-s + (−0.707 + 0.707i)32-s + (0.734 + 0.533i)35-s + ⋯
L(s)  = 1  + (0.437 − 0.437i)2-s + 0.618i·4-s + (−0.987 + 0.156i)5-s + (−0.642 − 0.642i)7-s + (0.707 + 0.707i)8-s + (−0.363 + 0.5i)10-s + (1.39 − 1.39i)13-s − 0.561·14-s + (−0.0966 − 0.610i)20-s + (1.34 + 1.34i)23-s + (0.951 − 0.309i)25-s − 1.22i·26-s + (0.396 − 0.396i)28-s + (−0.707 + 0.707i)32-s + (0.734 + 0.533i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2385\)    =    \(3^{2} \cdot 5 \cdot 53\)
Sign: $0.997 + 0.0746i$
Analytic conductor: \(1.19027\)
Root analytic conductor: \(1.09099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2385} (847, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2385,\ (\ :0),\ 0.997 + 0.0746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.304059417\)
\(L(\frac12)\) \(\approx\) \(1.304059417\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.987 - 0.156i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-0.437 + 0.437i)T - iT^{2} \)
7 \( 1 + (0.642 + 0.642i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1.34 - 1.34i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.26 - 1.26i)T + iT^{2} \)
41 \( 1 - 0.907iT - T^{2} \)
43 \( 1 + (-0.221 + 0.221i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.97iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.831 + 0.831i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
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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

−9.006423327748631158321514169512, −8.217492546346832642245623093330, −7.67995670649498011129855584756, −7.00622704132543432037759446800, −6.02280653992593419708723351926, −4.94773699718279036657986722436, −4.06663784184373585624504802437, −3.29241782299955602921405867327, −3.01985041157136213918619628359, −1.11453472166033040283013546191, 1.04171931491444531368818228823, 2.50661792629145290056386901608, 3.80268738593218170562265816483, 4.31184623655916761633751443989, 5.26514031119891564254918885592, 6.15814282881366378309211973075, 6.70384167375831558933503080206, 7.37609064779131334691810985231, 8.612248409596782427639781757661, 8.950361567851638159523468974729

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