Properties

Label 2-4015-1.1-c1-0-184
Degree $2$
Conductor $4015$
Sign $-1$
Analytic cond. $32.0599$
Root an. cond. $5.66214$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.96·2-s + 2.18·3-s + 1.87·4-s − 5-s − 4.30·6-s + 1.92·7-s + 0.237·8-s + 1.77·9-s + 1.96·10-s − 11-s + 4.10·12-s + 1.43·13-s − 3.79·14-s − 2.18·15-s − 4.22·16-s − 7.20·17-s − 3.49·18-s + 4.56·19-s − 1.87·20-s + 4.20·21-s + 1.96·22-s − 0.678·23-s + 0.518·24-s + 25-s − 2.83·26-s − 2.67·27-s + 3.62·28-s + ⋯
L(s)  = 1  − 1.39·2-s + 1.26·3-s + 0.939·4-s − 0.447·5-s − 1.75·6-s + 0.728·7-s + 0.0838·8-s + 0.591·9-s + 0.622·10-s − 0.301·11-s + 1.18·12-s + 0.398·13-s − 1.01·14-s − 0.564·15-s − 1.05·16-s − 1.74·17-s − 0.823·18-s + 1.04·19-s − 0.420·20-s + 0.918·21-s + 0.419·22-s − 0.141·23-s + 0.105·24-s + 0.200·25-s − 0.555·26-s − 0.515·27-s + 0.684·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4015\)    =    \(5 \cdot 11 \cdot 73\)
Sign: $-1$
Analytic conductor: \(32.0599\)
Root analytic conductor: \(5.66214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4015,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
11 \( 1 + T \)
73 \( 1 + T \)
good2 \( 1 + 1.96T + 2T^{2} \)
3 \( 1 - 2.18T + 3T^{2} \)
7 \( 1 - 1.92T + 7T^{2} \)
13 \( 1 - 1.43T + 13T^{2} \)
17 \( 1 + 7.20T + 17T^{2} \)
19 \( 1 - 4.56T + 19T^{2} \)
23 \( 1 + 0.678T + 23T^{2} \)
29 \( 1 + 8.50T + 29T^{2} \)
31 \( 1 + 0.232T + 31T^{2} \)
37 \( 1 - 9.23T + 37T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 - 1.35T + 43T^{2} \)
47 \( 1 + 3.95T + 47T^{2} \)
53 \( 1 + 5.34T + 53T^{2} \)
59 \( 1 - 1.88T + 59T^{2} \)
61 \( 1 + 5.04T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 6.04T + 71T^{2} \)
79 \( 1 + 0.0160T + 79T^{2} \)
83 \( 1 - 1.00T + 83T^{2} \)
89 \( 1 - 4.39T + 89T^{2} \)
97 \( 1 + 8.96T + 97T^{2} \)
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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

−8.317251093507205189748467530543, −7.58700861662377374622701937110, −7.25146352563343328185726568436, −6.13029689798385548338600336760, −4.88106092375139224981793831978, −4.14508355949049264217531295045, −3.14861696502191095339796451354, −2.19650848450830146315670018407, −1.47979579538124510919821356887, 0, 1.47979579538124510919821356887, 2.19650848450830146315670018407, 3.14861696502191095339796451354, 4.14508355949049264217531295045, 4.88106092375139224981793831978, 6.13029689798385548338600336760, 7.25146352563343328185726568436, 7.58700861662377374622701937110, 8.317251093507205189748467530543

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