Properties

Label 2-4015-1.1-c1-0-223
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.16·2-s − 0.343·3-s + 2.69·4-s + 5-s − 0.744·6-s − 0.845·7-s + 1.51·8-s − 2.88·9-s + 2.16·10-s − 11-s − 0.925·12-s − 2.48·13-s − 1.83·14-s − 0.343·15-s − 2.11·16-s + 2.71·17-s − 6.24·18-s − 2.40·19-s + 2.69·20-s + 0.290·21-s − 2.16·22-s + 1.41·23-s − 0.518·24-s + 25-s − 5.38·26-s + 2.01·27-s − 2.28·28-s + ⋯
L(s)  = 1  + 1.53·2-s − 0.198·3-s + 1.34·4-s + 0.447·5-s − 0.303·6-s − 0.319·7-s + 0.534·8-s − 0.960·9-s + 0.685·10-s − 0.301·11-s − 0.267·12-s − 0.688·13-s − 0.489·14-s − 0.0886·15-s − 0.529·16-s + 0.659·17-s − 1.47·18-s − 0.552·19-s + 0.603·20-s + 0.0633·21-s − 0.462·22-s + 0.294·23-s − 0.105·24-s + 0.200·25-s − 1.05·26-s + 0.388·27-s − 0.431·28-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 \( 1 - 2.16T + 2T^{2} \)
3 \( 1 + 0.343T + 3T^{2} \)
7 \( 1 + 0.845T + 7T^{2} \)
13 \( 1 + 2.48T + 13T^{2} \)
17 \( 1 - 2.71T + 17T^{2} \)
19 \( 1 + 2.40T + 19T^{2} \)
23 \( 1 - 1.41T + 23T^{2} \)
29 \( 1 + 4.89T + 29T^{2} \)
31 \( 1 - 0.199T + 31T^{2} \)
37 \( 1 + 7.56T + 37T^{2} \)
41 \( 1 - 4.58T + 41T^{2} \)
43 \( 1 + 2.75T + 43T^{2} \)
47 \( 1 - 2.01T + 47T^{2} \)
53 \( 1 + 7.78T + 53T^{2} \)
59 \( 1 + 7.00T + 59T^{2} \)
61 \( 1 + 1.78T + 61T^{2} \)
67 \( 1 + 1.37T + 67T^{2} \)
71 \( 1 - 5.24T + 71T^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 + 6.50T + 83T^{2} \)
89 \( 1 - 9.52T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.897359124323035182108283244511, −7.01842670595241231900275103657, −6.30359863395371419750133358162, −5.66213029051957887539339413782, −5.17682887147369367290215856017, −4.41161008132046622368083347394, −3.35696618127503052592723462395, −2.84243350084489133842692709657, −1.88716201883276430710685617052, 0, 1.88716201883276430710685617052, 2.84243350084489133842692709657, 3.35696618127503052592723462395, 4.41161008132046622368083347394, 5.17682887147369367290215856017, 5.66213029051957887539339413782, 6.30359863395371419750133358162, 7.01842670595241231900275103657, 7.897359124323035182108283244511

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