Properties

Label 2-8015-1.1-c1-0-268
Degree $2$
Conductor $8015$
Sign $-1$
Analytic cond. $64.0000$
Root an. cond. $8.00000$
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  + 0.938·2-s − 0.687·3-s − 1.11·4-s − 5-s − 0.645·6-s − 7-s − 2.92·8-s − 2.52·9-s − 0.938·10-s − 2.61·11-s + 0.769·12-s + 0.664·13-s − 0.938·14-s + 0.687·15-s − 0.509·16-s + 1.47·17-s − 2.37·18-s + 8.67·19-s + 1.11·20-s + 0.687·21-s − 2.45·22-s − 1.00·23-s + 2.01·24-s + 25-s + 0.623·26-s + 3.80·27-s + 1.11·28-s + ⋯
L(s)  = 1  + 0.663·2-s − 0.397·3-s − 0.559·4-s − 0.447·5-s − 0.263·6-s − 0.377·7-s − 1.03·8-s − 0.842·9-s − 0.296·10-s − 0.787·11-s + 0.222·12-s + 0.184·13-s − 0.250·14-s + 0.177·15-s − 0.127·16-s + 0.357·17-s − 0.559·18-s + 1.98·19-s + 0.250·20-s + 0.150·21-s − 0.522·22-s − 0.209·23-s + 0.411·24-s + 0.200·25-s + 0.122·26-s + 0.731·27-s + 0.211·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8015\)    =    \(5 \cdot 7 \cdot 229\)
Sign: $-1$
Analytic conductor: \(64.0000\)
Root analytic conductor: \(8.00000\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8015,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
229 \( 1 + T \)
good2 \( 1 - 0.938T + 2T^{2} \)
3 \( 1 + 0.687T + 3T^{2} \)
11 \( 1 + 2.61T + 11T^{2} \)
13 \( 1 - 0.664T + 13T^{2} \)
17 \( 1 - 1.47T + 17T^{2} \)
19 \( 1 - 8.67T + 19T^{2} \)
23 \( 1 + 1.00T + 23T^{2} \)
29 \( 1 - 6.73T + 29T^{2} \)
31 \( 1 + 3.63T + 31T^{2} \)
37 \( 1 + 1.50T + 37T^{2} \)
41 \( 1 - 3.10T + 41T^{2} \)
43 \( 1 + 7.59T + 43T^{2} \)
47 \( 1 + 4.17T + 47T^{2} \)
53 \( 1 + 8.29T + 53T^{2} \)
59 \( 1 - 13.1T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + 6.03T + 67T^{2} \)
71 \( 1 - 5.79T + 71T^{2} \)
73 \( 1 + 0.203T + 73T^{2} \)
79 \( 1 + 6.25T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + 6.42T + 89T^{2} \)
97 \( 1 - 3.37T + 97T^{2} \)
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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

−7.49232303197060222475854566958, −6.61322134459631580569401829919, −5.90294473633719736197507947616, −5.16353137778429101376781391852, −5.00663931307139504161691718383, −3.81382419139261779340200423637, −3.25300830684885667010759398296, −2.65057035120131589808141792628, −0.978250671128987457411758596898, 0, 0.978250671128987457411758596898, 2.65057035120131589808141792628, 3.25300830684885667010759398296, 3.81382419139261779340200423637, 5.00663931307139504161691718383, 5.16353137778429101376781391852, 5.90294473633719736197507947616, 6.61322134459631580569401829919, 7.49232303197060222475854566958

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