↑

Properties

Label 2-320-320.43-c1-0-39
Degree $2$
Conductor $320$
Sign $0.574 + 0.818i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
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  + (1.30 + 0.544i)2-s + (−1.17 − 0.783i)3-s + (1.40 + 1.42i)4-s + (0.215 − 2.22i)5-s + (−1.10 − 1.66i)6-s + (−1.53 − 3.71i)7-s + (1.06 + 2.62i)8-s + (−0.387 − 0.935i)9-s + (1.49 − 2.78i)10-s + (0.657 − 3.30i)11-s + (−0.537 − 2.76i)12-s + (0.810 + 4.07i)13-s + (0.0128 − 5.68i)14-s + (−1.99 + 2.44i)15-s + (−0.0361 + 3.99i)16-s + 4.22i·17-s + ⋯
L(s)  = 1  + (0.923 + 0.384i)2-s + (−0.676 − 0.452i)3-s + (0.703 + 0.710i)4-s + (0.0964 − 0.995i)5-s + (−0.450 − 0.677i)6-s + (−0.581 − 1.40i)7-s + (0.376 + 0.926i)8-s + (−0.129 − 0.311i)9-s + (0.472 − 0.881i)10-s + (0.198 − 0.996i)11-s + (−0.155 − 0.799i)12-s + (0.224 + 1.12i)13-s + (0.00342 − 1.51i)14-s + (−0.515 + 0.630i)15-s + (−0.00903 + 0.999i)16-s + 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52519 - 0.792359i\)
\(L(\frac12)\) \(\approx\) \(1.52519 - 0.792359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 - 0.544i)T \)
5 \( 1 + (-0.215 + 2.22i)T \)
good3 \( 1 + (1.17 + 0.783i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (1.53 + 3.71i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.657 + 3.30i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (-0.810 - 4.07i)T + (-12.0 + 4.97i)T^{2} \)
17 \( 1 - 4.22iT - 17T^{2} \)
19 \( 1 + (-3.18 + 4.77i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (-5.70 - 2.36i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.385 + 1.93i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + 3.07T + 31T^{2} \)
37 \( 1 + (-11.8 - 2.35i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (1.06 + 2.56i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-0.0702 - 0.105i)T + (-16.4 + 39.7i)T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + (-2.48 - 3.72i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-6.94 + 4.64i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-1.25 - 6.31i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-1.57 + 2.35i)T + (-25.6 - 61.8i)T^{2} \)
71 \( 1 + (4.67 - 11.2i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-4.01 - 1.66i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.55 - 3.55i)T + 79iT^{2} \)
83 \( 1 + (-0.407 - 2.05i)T + (-76.6 + 31.7i)T^{2} \)
89 \( 1 + (-2.63 - 1.08i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (8.50 + 8.50i)T + 97iT^{2} \)
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

−11.42486711617980869417152121713, −11.21474142460365763800599342955, −9.561119496810002801859676480347, −8.504273390822618700511592314583, −7.22695892243531643688568750501, −6.51478772258445817181872621391, −5.63725316028594008091194803628, −4.41696323642451647021278388822, −3.47245089106390565207736005158, −1.08796445104146339921521840160, 2.41309103627638934403930247411, 3.30307576084415134030722123380, 4.95424987215793990041552614137, 5.66327163065548223081175134302, 6.50041582400305709051607824964, 7.68224550315040981637726053084, 9.505546765900925956665943291437, 10.10520205062467508139289206229, 11.06948518475514048942998170562, 11.72152566399999434122832598206

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