L-function 1-1925-1925.1033-r0-0-0

• Label: 1-1925-1925.1033-r0-0-0
• Degree: $1$
• Conductor: $1925$
• Sign: $0.103 - 0.994i$
• Analytic cond.: $8.93966$
• Root an. cond.: $8.93966$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.994 − 0.104i)2^-s + (−0.207 − 0.978i)3^-s + (0.978 − 0.207i)4^-s + (−0.309 − 0.951i)6^-s + (0.951 − 0.309i)8^-s + (−0.913 + 0.406i)9^-s + (−0.406 − 0.913i)12^-s + (−0.587 + 0.809i)13^-s + (0.913 − 0.406i)16^-s + (0.743 + 0.669i)17^-s + (−0.866 + 0.5i)18^-s + (−0.978 − 0.207i)19^-s + (0.994 − 0.104i)23^-s + (−0.5 − 0.866i)24^-s + (−0.5 + 0.866i)26^-s + (0.587 + 0.809i)27^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1925$    =    $5^{2} \cdot 7 \cdot 11$
Sign:	$0.103 - 0.994i$
Analytic conductor:	$8.93966$
Root analytic conductor:	$8.93966$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1925} (1033, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1925,\ (0:\ ),\ 0.103 - 0.994i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.164386496 - 1.949887336i$
$L(1)$	$\approx$	$1.686199655 - 0.7351479766i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	7	$1$
	11	$1$
good	2	$1 + (0.994 - 0.104i)T$
	3	$1 + (-0.207 - 0.978i)T$
	13	$1 + (-0.587 + 0.809i)T$
	17	$1 + (0.743 + 0.669i)T$
	19	$1 + (-0.978 - 0.207i)T$
	23	$1 + (0.994 - 0.104i)T$
	29	$1 + (0.309 - 0.951i)T$
	31	$1 + (0.669 - 0.743i)T$
	37	$1 + (-0.406 - 0.913i)T$

Imaginary part of the first few zeros on the critical line

−20.53806031850539937815217403991, −19.74319420899418285608077445158, −18.99463621724670069196213517271, −17.67023259405825577288968033367, −17.07972430196664508751888039144, −16.379015939889594804868352240921, −15.6884626150529680468995135075, −15.03811428359157793929279621493, −14.461114717614812930135059180117, −13.77457618789232237882719067328, −12.6564488450134369508455237544, −12.23225164509372029696359350782, −11.283731230714105944816689931664, −10.60336892158621059427183616862, −10.03562490147658351129951520289, −8.9891041005939561018672654763, −8.07137642883569443017714125761, −7.15837749079534678568893520884, −6.26648443634155396035648418574, −5.41041326489041928883686294594, −4.88513455408097688694693903841, −4.12812932671802790783432033706, −3.072713324541942694174306557716, −2.7252919434405853086593908029, −1.14431321301914929427559972403, 0.79649706959103490840490806954, 1.979709995228307225986144333950, 2.44284503063718747404194193427, 3.57661537441738653281939759967, 4.50175240724650657140987058759, 5.34503544608967806790919775481, 6.18968497848497632093865723952, 6.74478451249524153927410621979, 7.536932174357441833159607599302, 8.28234451768059746560570057011, 9.407205785649753792070783799572, 10.52878468369261817241075769672, 11.14375506085641221218862735437, 12.06201892316098629410320954108, 12.380364520275232707319269105067, 13.2778514694882312317396790050, 13.77808336755639310811970203329, 14.68823592876277779396002280109, 15.09561469353460013160922677769, 16.2969307489255705687004235584, 16.96484718310403906675857722934, 17.46901626748918915069155289063, 18.74366971320351347953461143355, 19.28434553585982859258396964900, 19.634146331213860281713786508468

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