↑

Properties

Label 2-795-795.317-c1-0-40
Degree $2$
Conductor $795$
Sign $0.671 + 0.740i$
Analytic cond. $6.34810$
Root an. cond. $2.51954$
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  + (−1.96 − 1.96i)2-s + (−1.22 + 1.22i)3-s + 5.75i·4-s + (2.19 − 0.407i)5-s + 4.82·6-s + (2.79 + 2.79i)7-s + (7.40 − 7.40i)8-s − 2.99i·9-s + (−5.13 − 3.52i)10-s + (−7.05 − 7.05i)12-s + (2.99 − 2.99i)13-s − 11.0i·14-s + (−2.19 + 3.19i)15-s − 17.6·16-s + (−5.90 + 5.90i)18-s + ⋯
L(s)  = 1  + (−1.39 − 1.39i)2-s + (−0.707 + 0.707i)3-s + 2.87i·4-s + (0.983 − 0.182i)5-s + 1.96·6-s + (1.05 + 1.05i)7-s + (2.61 − 2.61i)8-s − 0.999i·9-s + (−1.62 − 1.11i)10-s + (−2.03 − 2.03i)12-s + (0.830 − 0.830i)13-s − 2.94i·14-s + (−0.566 + 0.824i)15-s − 4.41·16-s + (−1.39 + 1.39i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 795 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 795 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(795\)    =    \(3 \cdot 5 \cdot 53\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(6.34810\)
Root analytic conductor: \(2.51954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{795} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 795,\ (\ :1/2),\ 0.671 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.774428 - 0.343065i\)
\(L(\frac12)\) \(\approx\) \(0.774428 - 0.343065i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (-2.19 + 0.407i)T \)
53 \( 1 + (5.14 - 5.14i)T \)
good2 \( 1 + (1.96 + 1.96i)T + 2iT^{2} \)
7 \( 1 + (-2.79 - 2.79i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-2.99 + 2.99i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-5.77 + 5.77i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (1.49 + 1.49i)T + 37iT^{2} \)
41 \( 1 + 12.5T + 41T^{2} \)
43 \( 1 + (-9.25 + 9.25i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (3.82 - 3.82i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (2.13 + 2.13i)T + 97iT^{2} \)
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

−10.38192703295747532977987085046, −9.375480036526417497994142111436, −8.783155525506894648396727247957, −8.279673247767631331368718243264, −6.82133221235683307976495918379, −5.55897724722123650990662801087, −4.62676333199790022771267638814, −3.25193035995017650856536518044, −2.17290147560793026608991787649, −0.979903105153824375920522476529, 1.15663843681658856654091672257, 1.72601178985367398772267599991, 4.75350874062747892130796910424, 5.44437423886495353094326700670, 6.41351737528901149545213756059, 6.94567637825096787959749063531, 7.67575185371860102681664003656, 8.484284096286942593615555041851, 9.400824635669439769989114888207, 10.25616292699485846769212155764

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