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Properties

Label 1-605-605.142-r0-0-0
Degree $1$
Conductor $605$
Sign $-0.137 + 0.990i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.755 + 0.654i)2-s − i·3-s + (0.142 + 0.989i)4-s + (0.654 − 0.755i)6-s + (−0.281 + 0.959i)7-s + (−0.540 + 0.841i)8-s − 9-s + (0.989 − 0.142i)12-s + (0.989 − 0.142i)13-s + (−0.841 + 0.540i)14-s + (−0.959 + 0.281i)16-s + (−0.909 − 0.415i)17-s + (−0.755 − 0.654i)18-s + (0.415 + 0.909i)19-s + (0.959 + 0.281i)21-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)2-s − i·3-s + (0.142 + 0.989i)4-s + (0.654 − 0.755i)6-s + (−0.281 + 0.959i)7-s + (−0.540 + 0.841i)8-s − 9-s + (0.989 − 0.142i)12-s + (0.989 − 0.142i)13-s + (−0.841 + 0.540i)14-s + (−0.959 + 0.281i)16-s + (−0.909 − 0.415i)17-s + (−0.755 − 0.654i)18-s + (0.415 + 0.909i)19-s + (0.959 + 0.281i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.137 + 0.990i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ -0.137 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.136457163 + 1.305406438i\)
\(L(\frac12)\) \(\approx\) \(1.136457163 + 1.305406438i\)
\(L(1)\) \(\approx\) \(1.301995994 + 0.5351321073i\)
\(L(1)\) \(\approx\) \(1.301995994 + 0.5351321073i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.755 + 0.654i)T \)
3 \( 1 - iT \)
7 \( 1 + (-0.281 + 0.959i)T \)
13 \( 1 + (0.989 - 0.142i)T \)
17 \( 1 + (-0.909 - 0.415i)T \)
19 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (0.281 + 0.959i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (-0.989 - 0.142i)T \)
41 \( 1 + (0.654 - 0.755i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (0.755 - 0.654i)T \)
53 \( 1 + (-0.281 + 0.959i)T \)
59 \( 1 + (0.654 + 0.755i)T \)
61 \( 1 + (0.654 + 0.755i)T \)
67 \( 1 + (-0.755 - 0.654i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (-0.281 - 0.959i)T \)
79 \( 1 + (0.841 - 0.540i)T \)
83 \( 1 + (0.281 - 0.959i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (-0.540 + 0.841i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.53770332295471544508211882183, −22.19777020219931464464862153195, −21.03581387379835972566267015895, −20.604520161456042970255724398483, −19.860624910534627210552323451721, −19.083437181826519265739617329019, −17.83431857755578640381628427918, −16.87053238165930793877296405595, −15.870869294681203760678368936722, −15.36085551691530772979173983116, −14.28671256668808214436990737668, −13.60260318330801923340423182426, −12.87443662045713494154579401089, −11.488798418079316197337434675, −10.99438955352653469722591957147, −10.2318523822154759588504649343, −9.407567581471308234160549434905, −8.43358490675853212324822959011, −6.820014507198516527609068717400, −6.03892065044623926776671712577, −4.83415247442830311050609670313, −4.146780393364449973856671084580, −3.41470761784125008478345266145, −2.30509967670334422176986133791, −0.67617804240070323693090510120, 1.61724309609093192013261832194, 2.78281164364874654357788057767, 3.60677448870363274998072831444, 5.16717741656638537388090579535, 5.83603350182759165384512516082, 6.65417963247197045983588178303, 7.47208576335812224108457401080, 8.546533113390347033013296378140, 9.03513742309058535379707697543, 10.86630413894186681885163772117, 11.85533448597701855662182300618, 12.39183105784558729531228456249, 13.302940636821338685730223779099, 13.88926689108985822534533658561, 14.84826318268206183690070390882, 15.77572721663192872264210532561, 16.39385670006459538702383426712, 17.7055299924924606153902721207, 18.06750596589944824512646771640, 19.03711264981682113595408664867, 20.05219033326546030084624212803, 20.95870940291292901915508865290, 21.890581254069256633254184449357, 22.715110525328600693081020017939, 23.29938936232914959722654382421

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