↑

Properties

Label 2-775-5.4-c1-0-39
Degree $2$
Conductor $775$
Sign $-0.447 + 0.894i$
Analytic cond. $6.18840$
Root an. cond. $2.48765$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.15i·2-s − 3.02i·3-s + 0.660·4-s + 3.50·6-s − 1.50i·7-s + 3.07i·8-s − 6.16·9-s − 5.81·11-s − 2.00i·12-s − 5.50i·13-s + 1.73·14-s − 2.24·16-s − 0.790i·17-s − 7.13i·18-s + 3.89·19-s + ⋯
L(s)  = 1  + 0.818i·2-s − 1.74i·3-s + 0.330·4-s + 1.43·6-s − 0.568i·7-s + 1.08i·8-s − 2.05·9-s − 1.75·11-s − 0.577i·12-s − 1.52i·13-s + 0.464·14-s − 0.560·16-s − 0.191i·17-s − 1.68i·18-s + 0.894·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640917 - 1.03702i\)
\(L(\frac12)\) \(\approx\) \(0.640917 - 1.03702i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.15iT - 2T^{2} \)
3 \( 1 + 3.02iT - 3T^{2} \)
7 \( 1 + 1.50iT - 7T^{2} \)
11 \( 1 + 5.81T + 11T^{2} \)
13 \( 1 + 5.50iT - 13T^{2} \)
17 \( 1 + 0.790iT - 17T^{2} \)
19 \( 1 - 3.89T + 19T^{2} \)
23 \( 1 + 5.24iT - 23T^{2} \)
29 \( 1 + 6.55T + 29T^{2} \)
37 \( 1 - 0.476iT - 37T^{2} \)
41 \( 1 + 3.40T + 41T^{2} \)
43 \( 1 + 2.79iT - 43T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 - 4.13T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 - 1.81iT - 67T^{2} \)
71 \( 1 - 5.16T + 71T^{2} \)
73 \( 1 + 6.45iT - 73T^{2} \)
79 \( 1 + 0.510T + 79T^{2} \)
83 \( 1 + 3.53iT - 83T^{2} \)
89 \( 1 + 1.12T + 89T^{2} \)
97 \( 1 - 0.260iT - 97T^{2} \)
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

−10.25832228905804189379925803211, −8.526294463308254516858754798924, −7.915748092906061357412730172696, −7.44183713699713073319785296490, −6.83162466983733098568760446966, −5.68637645539274740868843585142, −5.29980211964654597205086331622, −3.02829593131586919398233486246, −2.21267277312085705196321645591, −0.56229697365266261211405461604, 2.15327720344052714960060373154, 3.14104322615336554615238245385, 3.98711145772594691697423676011, 5.06966851755452923352532564506, 5.77648540596523734303848909727, 7.17706498766212568024449671941, 8.329316804672310787681337754431, 9.528221530990804665494334379004, 9.654334703263050106061970865319, 10.62737461218143838970975729142

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