↑

Properties

Label 2-308-28.19-c1-0-0
Degree $2$
Conductor $308$
Sign $0.0413 - 0.999i$
Analytic cond. $2.45939$
Root an. cond. $1.56824$
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  + (−0.216 − 1.39i)2-s + (−0.275 − 0.477i)3-s + (−1.90 + 0.604i)4-s + (−2.90 − 1.67i)5-s + (−0.608 + 0.488i)6-s + (−1.79 + 1.94i)7-s + (1.25 + 2.53i)8-s + (1.34 − 2.33i)9-s + (−1.71 + 4.42i)10-s + (0.866 − 0.5i)11-s + (0.814 + 0.744i)12-s + 6.84i·13-s + (3.10 + 2.08i)14-s + 1.84i·15-s + (3.27 − 2.30i)16-s + (−1.60 + 0.929i)17-s + ⋯
L(s)  = 1  + (−0.152 − 0.988i)2-s + (−0.159 − 0.275i)3-s + (−0.953 + 0.302i)4-s + (−1.29 − 0.749i)5-s + (−0.248 + 0.199i)6-s + (−0.678 + 0.735i)7-s + (0.444 + 0.895i)8-s + (0.449 − 0.778i)9-s + (−0.542 + 1.39i)10-s + (0.261 − 0.150i)11-s + (0.235 + 0.214i)12-s + 1.89i·13-s + (0.830 + 0.557i)14-s + 0.477i·15-s + (0.817 − 0.575i)16-s + (−0.390 + 0.225i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0413 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0413 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.0413 - 0.999i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.0413 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0446334 + 0.0428229i\)
\(L(\frac12)\) \(\approx\) \(0.0446334 + 0.0428229i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.216 + 1.39i)T \)
7 \( 1 + (1.79 - 1.94i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good3 \( 1 + (0.275 + 0.477i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.90 + 1.67i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 6.84iT - 13T^{2} \)
17 \( 1 + (1.60 - 0.929i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.24 - 3.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.38 + 3.10i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.62T + 29T^{2} \)
31 \( 1 + (3.18 + 5.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.10 - 3.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.408iT - 41T^{2} \)
43 \( 1 + 5.67iT - 43T^{2} \)
47 \( 1 + (-1.45 + 2.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.42 - 2.47i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.59 - 9.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.28 - 1.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.675 - 0.389i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.98iT - 71T^{2} \)
73 \( 1 + (7.78 - 4.49i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.96 + 4.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.77T + 83T^{2} \)
89 \( 1 + (7.66 + 4.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.7iT - 97T^{2} \)
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

−11.89127031529744377193140957060, −11.47930460445738829907088206096, −9.993187389772032411374808742361, −9.043585583428301071796781059044, −8.551027079725063807057641427567, −7.23727036319104895163713753325, −5.96595387895306976722099565080, −4.20857653409180949401029458763, −3.85529368605531462714604645354, −1.87757709608847241977797006471, 0.04821177450064808805563742577, 3.41427946642534256905403924241, 4.29857484038548121261948036634, 5.55347099802387591652972294111, 6.92518325841022736489363220974, 7.48746606647624394341466150983, 8.228314647417692896833934372765, 9.656820855741500812260638293327, 10.52158526538707635914774844158, 11.11979343711677289972773243796

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