↑

Properties

Label 2-368-16.5-c1-0-36
Degree $2$
Conductor $368$
Sign $0.874 + 0.485i$
Analytic cond. $2.93849$
Root an. cond. $1.71420$
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  + (1.29 + 0.578i)2-s + (1.73 − 1.73i)3-s + (1.33 + 1.49i)4-s + (−1.86 − 1.86i)5-s + (3.23 − 1.23i)6-s − 1.07i·7-s + (0.855 + 2.69i)8-s − 2.98i·9-s + (−1.33 − 3.49i)10-s + (0.551 + 0.551i)11-s + (4.88 + 0.278i)12-s + (0.244 − 0.244i)13-s + (0.621 − 1.38i)14-s − 6.46·15-s + (−0.454 + 3.97i)16-s + 3.95·17-s + ⋯
L(s)  = 1  + (0.912 + 0.408i)2-s + (0.998 − 0.998i)3-s + (0.665 + 0.746i)4-s + (−0.835 − 0.835i)5-s + (1.31 − 0.503i)6-s − 0.406i·7-s + (0.302 + 0.953i)8-s − 0.995i·9-s + (−0.421 − 1.10i)10-s + (0.166 + 0.166i)11-s + (1.41 + 0.0803i)12-s + (0.0676 − 0.0676i)13-s + (0.166 − 0.370i)14-s − 1.66·15-s + (−0.113 + 0.993i)16-s + 0.958·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.874 + 0.485i$
Analytic conductor: \(2.93849\)
Root analytic conductor: \(1.71420\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :1/2),\ 0.874 + 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.61906 - 0.677949i\)
\(L(\frac12)\) \(\approx\) \(2.61906 - 0.677949i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.578i)T \)
23 \( 1 - iT \)
good3 \( 1 + (-1.73 + 1.73i)T - 3iT^{2} \)
5 \( 1 + (1.86 + 1.86i)T + 5iT^{2} \)
7 \( 1 + 1.07iT - 7T^{2} \)
11 \( 1 + (-0.551 - 0.551i)T + 11iT^{2} \)
13 \( 1 + (-0.244 + 0.244i)T - 13iT^{2} \)
17 \( 1 - 3.95T + 17T^{2} \)
19 \( 1 + (2.75 - 2.75i)T - 19iT^{2} \)
29 \( 1 + (4.25 - 4.25i)T - 29iT^{2} \)
31 \( 1 + 2.33T + 31T^{2} \)
37 \( 1 + (3.31 + 3.31i)T + 37iT^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
43 \( 1 + (8.16 + 8.16i)T + 43iT^{2} \)
47 \( 1 - 4.08T + 47T^{2} \)
53 \( 1 + (0.106 + 0.106i)T + 53iT^{2} \)
59 \( 1 + (-4.32 - 4.32i)T + 59iT^{2} \)
61 \( 1 + (6.39 - 6.39i)T - 61iT^{2} \)
67 \( 1 + (-3.60 + 3.60i)T - 67iT^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + 6.16iT - 73T^{2} \)
79 \( 1 - 1.42T + 79T^{2} \)
83 \( 1 + (-4.07 + 4.07i)T - 83iT^{2} \)
89 \( 1 - 9.40iT - 89T^{2} \)
97 \( 1 - 3.27T + 97T^{2} \)
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

−11.93529625786628450076115924882, −10.62435062296244368061705622020, −9.006729456468286289542942586833, −8.146776695650649158322614312436, −7.63128574783214032735802959407, −6.78810429184572225064337265093, −5.42531955924970885872087722242, −4.15538647809956733942896790948, −3.25722020181861591151610196326, −1.66368358065988289861878393599, 2.46409238528425893484450480780, 3.45202737014210836014899898999, 4.04828664753916538799321924118, 5.30803494397197178751393828404, 6.63984815454228368133162742588, 7.72911690080378929714869152598, 8.865295899851648455190663147565, 9.846524486673896234426364847271, 10.65520891812388269672909710833, 11.43192635605095202040183707277

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