↑

Properties

Label 1-712-712.605-r0-0-0
Degree $1$
Conductor $712$
Sign $0.886 + 0.462i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.281 + 0.959i)3-s + (0.415 − 0.909i)5-s + (0.909 + 0.415i)7-s + (−0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.281 − 0.959i)13-s + (0.989 + 0.142i)15-s + (0.142 + 0.989i)17-s + (0.540 + 0.841i)19-s + (−0.142 + 0.989i)21-s + (0.540 + 0.841i)23-s + (−0.654 − 0.755i)25-s + (−0.755 − 0.654i)27-s + (0.909 + 0.415i)29-s + (0.540 − 0.841i)31-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)3-s + (0.415 − 0.909i)5-s + (0.909 + 0.415i)7-s + (−0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.281 − 0.959i)13-s + (0.989 + 0.142i)15-s + (0.142 + 0.989i)17-s + (0.540 + 0.841i)19-s + (−0.142 + 0.989i)21-s + (0.540 + 0.841i)23-s + (−0.654 − 0.755i)25-s + (−0.755 − 0.654i)27-s + (0.909 + 0.415i)29-s + (0.540 − 0.841i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.886 + 0.462i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (605, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.886 + 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.784354822 + 0.4377184186i\)
\(L(\frac12)\) \(\approx\) \(1.784354822 + 0.4377184186i\)
\(L(1)\) \(\approx\) \(1.324629539 + 0.2288938309i\)
\(L(1)\) \(\approx\) \(1.324629539 + 0.2288938309i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.281 + 0.959i)T \)
5 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (0.909 + 0.415i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (-0.281 - 0.959i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (0.540 + 0.841i)T \)
23 \( 1 + (0.540 + 0.841i)T \)
29 \( 1 + (0.909 + 0.415i)T \)
31 \( 1 + (0.540 - 0.841i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.281 - 0.959i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + (0.959 + 0.281i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (0.281 - 0.959i)T \)
61 \( 1 + (0.755 + 0.654i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (-0.989 + 0.142i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.87477887892980206256859004504, −21.62934385322616692355384079734, −20.86660612739628620473681151590, −20.06950142818537169177602430425, −19.16534640331302905395944352364, −18.36579730647140871933633087254, −17.76619208971904276343947759338, −17.26165617433047669708985628333, −15.901616929370659119724021160387, −14.77567323114295591492247686818, −14.20030551753406322927524838869, −13.69737793979469660455617296012, −12.6142039057470732690397186383, −11.66674968676235184516194446324, −11.023014306242334866781599020132, −9.92016697554175932883466776395, −9.016234022486758215342460880686, −7.86037147083015037778003390633, −7.09286020560965245155138687153, −6.695698365813735933002084237245, −5.31511684469269536776286417557, −4.36386212375222739724444241046, −2.78605267384566331040506163598, −2.28406881250685995087697265460, −1.09575232552059193295918715588, 1.12102558544202672872161023275, 2.41561291175721872281911914303, 3.48453296521848614679640506590, 4.5683518021789519504971178599, 5.47145696841503840767036066886, 5.80577052594197602662867218930, 7.897352035301567672507805628818, 8.28917979689671653996898120875, 9.127882964939189226774150658, 10.08929062262629779000413113955, 10.80508162460902416965162911972, 11.788710670059333477996279003682, 12.73343480988708833276594416407, 13.75032328941886727555958648880, 14.41876720763583231726508465991, 15.46399494327968373345057309150, 15.91308327412187477375135171289, 17.15162731646875581353131195497, 17.33877859978051165250395588709, 18.64587529220910483799329550334, 19.59142410480056588590919126983, 20.588517153558469474847340549, 20.94744487614588792523183338797, 21.68681140180993525714198293784, 22.35224439566668972676982883891

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