L-function 1-1925-1925.303-r0-0-0

• Label: 1-1925-1925.303-r0-0-0
• Degree: $1$
• Conductor: $1925$
• Sign: $-0.386 - 0.922i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1925$    =    $5^{2} \cdot 7 \cdot 11$
Sign:	$-0.386 - 0.922i$
Analytic conductor:	$8.93966$
Root analytic conductor:	$8.93966$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1925} (303, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1925,\ (0:\ ),\ -0.386 - 0.922i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.113302926 - 3.175313940i$
$L(1)$	$\approx$	$1.980813099 - 1.189065224i$

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	$1 + (0.866 - 0.5i)T$
	3	$1 + (0.994 - 0.104i)T$
	13	$1 + (-0.951 - 0.309i)T$
	17	$1 + (0.994 - 0.104i)T$
	19	$1 + (-0.5 - 0.866i)T$
	23	$1 + (-0.994 - 0.104i)T$
	29	$1 + T$
	31	$1 + (-0.978 - 0.207i)T$
	37	$1 + (-0.743 - 0.669i)T$

Imaginary part of the first few zeros on the critical line

−20.498080735370223978822545487105, −19.59946071107772431973369801636, −19.05920647104695386337102543194, −18.01904364056201138967786281882, −17.15242011758578269091205544751, −16.305611934676452271635827291582, −15.83725353004856081565869710737, −14.85187319167129094706363937120, −14.35162431871867434365407231468, −14.003666837247997714522725854875, −12.888707532098890140912251961357, −12.43322525320362765780284353294, −11.6768729735465503679736574768, −10.440110676323266297321157475477, −9.80403259598101359323695353002, −8.77561304521173385401178501911, −8.010748984956787858458777238471, −7.480167643272952765787591174488, −6.63068229236851774602883268958, −5.6935791585735925680923982794, −4.77882551344351267899560960383, −4.01449293376661680892510284491, −3.30518270708054611927279335861, −2.42900746279573460284890798683, −1.63042131953126801412245641724, 0.7985888477466818502317210562, 2.01973848324662910735600721072, 2.56519606279200296005030673656, 3.442411077227497856762161299732, 4.193969110936550803258411847061, 5.02845487574828139763068342529, 5.91405828594005153320400509688, 6.99979426292182529709476808215, 7.52479698029807158848292611934, 8.583175877284796382521414013197, 9.47841158950830939773233887420, 10.13885716915753564204231092086, 10.79599425525761460767254666240, 12.05771626203213237659402702601, 12.3782505124920652875145663565, 13.242400442378202054074477654689, 13.95657128099786225575405130745, 14.51464725768895319277984401923, 15.15793801679648119589540484232, 15.82500334701423062320402968974, 16.69955869080645755988865433708, 17.86948029961029873147166801777, 18.61959910611409341666383303736, 19.46015961772563078713919382238, 19.818165170249102047060750202834

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