↑

Properties

Label 2-104-104.77-c1-0-2
Degree $2$
Conductor $104$
Sign $0.980 - 0.196i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
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 − i)2-s + i·3-s + 2i·4-s + 5-s + (1 − i)6-s + 3i·7-s + (2 − 2i)8-s + 2·9-s + (−1 − i)10-s + 2·11-s − 2·12-s + (−3 − 2i)13-s + (3 − 3i)14-s + i·15-s − 4·16-s + 3·17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 0.577i·3-s + i·4-s + 0.447·5-s + (0.408 − 0.408i)6-s + 1.13i·7-s + (0.707 − 0.707i)8-s + 0.666·9-s + (−0.316 − 0.316i)10-s + 0.603·11-s − 0.577·12-s + (−0.832 − 0.554i)13-s + (0.801 − 0.801i)14-s + 0.258i·15-s − 16-s + 0.727·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.980 - 0.196i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802096 + 0.0794232i\)
\(L(\frac12)\) \(\approx\) \(0.802096 + 0.0794232i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
13 \( 1 + (3 + 2i)T \)
good3 \( 1 - iT - 3T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 + 9iT - 43T^{2} \)
47 \( 1 + 7iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 5iT - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 - 4iT - 89T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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

−13.65256715401797432042833844266, −12.35409975707263402349255573602, −11.83154549957557831819446561218, −10.25760428819074643973910806183, −9.756467649995391558827101569606, −8.742668970694198924173312210402, −7.43213155263816955805373058027, −5.66492165023256605873583598016, −3.97335010471256464363848913478, −2.23083718496778732570938837681, 1.50989599007397932630514486544, 4.45607701323885764529889498638, 6.19223577176150942201911223395, 7.15030621521244837397206939435, 7.948701491373206449920631763674, 9.613508474892148105860213063569, 10.12002502784925675527084533184, 11.55168050213096256500942544377, 12.93652707664178297812045231949, 14.03138070170820377342404104863

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