L-function 1-520-520.123-r1-0-0

• Label: 1-520-520.123-r1-0-0
• Degree: $1$
• Conductor: $520$
• Sign: $-0.558 - 0.829i$
• Analytic cond.: $55.8817$
• Root an. cond.: $55.8817$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.866 − 0.5i)3^-s + (0.5 − 0.866i)7^-s + (0.5 − 0.866i)9^-s + (0.866 − 0.5i)11^-s + (−0.866 − 0.5i)17^-s + (−0.866 − 0.5i)19^-s − i·21^-s + (0.866 − 0.5i)23^-s − i·27^-s + (−0.5 − 0.866i)29^-s − i·31^-s + (0.5 − 0.866i)33^-s + (0.5 + 0.866i)37^-s + (−0.866 + 0.5i)41^-s + (0.866 + 0.5i)43^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$520$    =    $2^{3} \cdot 5 \cdot 13$
Sign:	$-0.558 - 0.829i$
Analytic conductor:	$55.8817$
Root analytic conductor:	$55.8817$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{520} (123, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 520,\ (1:\ ),\ -0.558 - 0.829i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.304245432 - 2.449312021i$
$L(1)$	$\approx$	$1.361780025 - 0.6730081530i$

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	13	$1$
good	3	$1 + (0.866 - 0.5i)T$
	7	$1 + (0.5 - 0.866i)T$
	11	$1 + (0.866 - 0.5i)T$
	17	$1 + (-0.866 - 0.5i)T$
	19	$1 + (-0.866 - 0.5i)T$
	23	$1 + (0.866 - 0.5i)T$
	29	$1 + (-0.5 - 0.866i)T$
	31	$1 - iT$
	37	$1 + (0.5 + 0.866i)T$

Imaginary part of the first few zeros on the critical line

−23.77359292232714143391574479333, −22.47608031297158998412574581503, −21.87104390891581776823788836199, −21.07952377120527745024516442490, −20.32422612919653750547107695360, −19.424783340916188700753798740012, −18.7802324071622867830215083686, −17.66965139807428601327834298745, −16.81221738102755126131191424012, −15.70825616872633954307717677873, −14.87896287422200218061234238798, −14.592613054572653508695733349680, −13.33337002454861408243848402183, −12.515685720272792376105825096335, −11.394575449690213985595018088110, −10.53739678565581362571974781392, −9.31372870782373480492577463355, −8.883107571663334770655310029838, −7.934657487909729686197701748027, −6.84443320574512924177904016608, −5.60539430132170016942202624114, −4.51183020652589788230474644098, −3.70924313875084204398289701920, −2.392133693151336590880811926859, −1.64421771867151098377724345601, 0.59835288864972498728627405, 1.63238358796977166125121417296, 2.81410085901913442771566811312, 3.91220440458630257758152297284, 4.75849754803225631675786883651, 6.48938782912598550307571335914, 6.97754550021958572288958925842, 8.14148170779946202285743992436, 8.803002782075202732023899984649, 9.76637068351208417972823309719, 10.961028110345439342770049776818, 11.72194347269559904361607803452, 13.03066873700883512931547161340, 13.52785688997862777284323689226, 14.47543608352016542804832212444, 15.029139534410500014680203874241, 16.26616878000587117145891654658, 17.2517748943842509414996201549, 17.93240619179196208250668603963, 19.06601943211990563575831279823, 19.62197123437172493098742312002, 20.45815929694562080861775379843, 21.12618566966737446414625215495, 22.16846225197706753199543532114, 23.23200664066446945281065867306

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