Properties

Label 2-4016-1.1-c1-0-42
Degree $2$
Conductor $4016$
Sign $1$
Analytic cond. $32.0679$
Root an. cond. $5.66285$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.55·3-s + 2.96·5-s − 0.820·7-s + 3.51·9-s + 3.75·11-s − 2.63·13-s − 7.56·15-s + 5.32·17-s + 5.04·19-s + 2.09·21-s − 2.33·23-s + 3.78·25-s − 1.31·27-s + 0.0387·29-s + 4.75·31-s − 9.58·33-s − 2.43·35-s − 1.34·37-s + 6.73·39-s + 0.422·41-s − 11.8·43-s + 10.4·45-s + 12.0·47-s − 6.32·49-s − 13.5·51-s + 2.90·53-s + 11.1·55-s + ⋯
L(s)  = 1  − 1.47·3-s + 1.32·5-s − 0.310·7-s + 1.17·9-s + 1.13·11-s − 0.731·13-s − 1.95·15-s + 1.29·17-s + 1.15·19-s + 0.456·21-s − 0.487·23-s + 0.757·25-s − 0.252·27-s + 0.00719·29-s + 0.853·31-s − 1.66·33-s − 0.411·35-s − 0.221·37-s + 1.07·39-s + 0.0659·41-s − 1.80·43-s + 1.55·45-s + 1.76·47-s − 0.903·49-s − 1.90·51-s + 0.398·53-s + 1.50·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4016\)    =    \(2^{4} \cdot 251\)
Sign: $1$
Analytic conductor: \(32.0679\)
Root analytic conductor: \(5.66285\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.574168820\)
\(L(\frac12)\) \(\approx\) \(1.574168820\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 + T \)
good3 \( 1 + 2.55T + 3T^{2} \)
5 \( 1 - 2.96T + 5T^{2} \)
7 \( 1 + 0.820T + 7T^{2} \)
11 \( 1 - 3.75T + 11T^{2} \)
13 \( 1 + 2.63T + 13T^{2} \)
17 \( 1 - 5.32T + 17T^{2} \)
19 \( 1 - 5.04T + 19T^{2} \)
23 \( 1 + 2.33T + 23T^{2} \)
29 \( 1 - 0.0387T + 29T^{2} \)
31 \( 1 - 4.75T + 31T^{2} \)
37 \( 1 + 1.34T + 37T^{2} \)
41 \( 1 - 0.422T + 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 - 2.90T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 - 1.76T + 61T^{2} \)
67 \( 1 + 3.46T + 67T^{2} \)
71 \( 1 - 6.11T + 71T^{2} \)
73 \( 1 - 2.52T + 73T^{2} \)
79 \( 1 + 0.164T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 - 4.30T + 97T^{2} \)
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

−8.546834072961169360838437297350, −7.40240121695098463017478235373, −6.72489665636114260609351510173, −6.14837416134540629930368224226, −5.48519668456024469597139177691, −5.10940061306206229946275857761, −3.99802004910107158420163397394, −2.89931760916935508985430989218, −1.66209411391971203035673562718, −0.827569532234503246548533699865, 0.827569532234503246548533699865, 1.66209411391971203035673562718, 2.89931760916935508985430989218, 3.99802004910107158420163397394, 5.10940061306206229946275857761, 5.48519668456024469597139177691, 6.14837416134540629930368224226, 6.72489665636114260609351510173, 7.40240121695098463017478235373, 8.546834072961169360838437297350

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