Properties

Label 2-775-1.1-c1-0-42
Degree $2$
Conductor $775$
Sign $-1$
Analytic cond. $6.18840$
Root an. cond. $2.48765$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.19·2-s − 2.27·3-s + 2.80·4-s − 4.99·6-s − 1.38·7-s + 1.75·8-s + 2.19·9-s − 4.99·11-s − 6.38·12-s − 2.43·13-s − 3.04·14-s − 1.75·16-s − 4.71·17-s + 4.80·18-s + 5.74·19-s + 3.16·21-s − 10.9·22-s + 1.38·23-s − 3.99·24-s − 5.33·26-s + 1.84·27-s − 3.89·28-s − 5.16·29-s − 31-s − 7.35·32-s + 11.3·33-s − 10.3·34-s + ⋯
L(s)  = 1  + 1.54·2-s − 1.31·3-s + 1.40·4-s − 2.03·6-s − 0.525·7-s + 0.620·8-s + 0.730·9-s − 1.50·11-s − 1.84·12-s − 0.675·13-s − 0.813·14-s − 0.438·16-s − 1.14·17-s + 1.13·18-s + 1.31·19-s + 0.691·21-s − 2.33·22-s + 0.289·23-s − 0.816·24-s − 1.04·26-s + 0.354·27-s − 0.735·28-s − 0.959·29-s − 0.179·31-s − 1.30·32-s + 1.98·33-s − 1.77·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-1$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ -1)\)

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 \( 1 \)
31 \( 1 + T \)
good2 \( 1 - 2.19T + 2T^{2} \)
3 \( 1 + 2.27T + 3T^{2} \)
7 \( 1 + 1.38T + 7T^{2} \)
11 \( 1 + 4.99T + 11T^{2} \)
13 \( 1 + 2.43T + 13T^{2} \)
17 \( 1 + 4.71T + 17T^{2} \)
19 \( 1 - 5.74T + 19T^{2} \)
23 \( 1 - 1.38T + 23T^{2} \)
29 \( 1 + 5.16T + 29T^{2} \)
37 \( 1 - 4.31T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 + 2.15T + 43T^{2} \)
47 \( 1 - 8.20T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 - 0.672T + 59T^{2} \)
61 \( 1 - 1.44T + 61T^{2} \)
67 \( 1 - 14.9T + 67T^{2} \)
71 \( 1 + 2.80T + 71T^{2} \)
73 \( 1 + 2.85T + 73T^{2} \)
79 \( 1 - 6.85T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 - 8.15T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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

−10.27699085661625832425720126960, −9.260876176320719473397778302939, −7.71356823351601381409413519694, −6.80439540217213358486665127649, −6.06984368134385137104138961596, −5.09037362219977112743454119641, −4.95584556129276486877954607778, −3.49178735540773958891886497528, −2.43716029150059304967424314359, 0, 2.43716029150059304967424314359, 3.49178735540773958891886497528, 4.95584556129276486877954607778, 5.09037362219977112743454119641, 6.06984368134385137104138961596, 6.80439540217213358486665127649, 7.71356823351601381409413519694, 9.260876176320719473397778302939, 10.27699085661625832425720126960

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