↑

Properties

Label 1-373-373.332-r0-0-0
Degree $1$
Conductor $373$
Sign $0.260 + 0.965i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (−0.151 − 0.988i)2-s + (−0.0506 − 0.998i)3-s + (−0.954 + 0.299i)4-s + (−0.528 − 0.848i)5-s + (−0.979 + 0.201i)6-s + (−0.250 − 0.968i)7-s + (0.440 + 0.897i)8-s + (−0.994 + 0.101i)9-s + (−0.758 + 0.651i)10-s + (0.440 − 0.897i)11-s + (0.347 + 0.937i)12-s + (−0.440 + 0.897i)13-s + (−0.918 + 0.394i)14-s + (−0.820 + 0.571i)15-s + (0.820 − 0.571i)16-s + (−0.954 + 0.299i)17-s + ⋯
L(s)  = 1  + (−0.151 − 0.988i)2-s + (−0.0506 − 0.998i)3-s + (−0.954 + 0.299i)4-s + (−0.528 − 0.848i)5-s + (−0.979 + 0.201i)6-s + (−0.250 − 0.968i)7-s + (0.440 + 0.897i)8-s + (−0.994 + 0.101i)9-s + (−0.758 + 0.651i)10-s + (0.440 − 0.897i)11-s + (0.347 + 0.937i)12-s + (−0.440 + 0.897i)13-s + (−0.918 + 0.394i)14-s + (−0.820 + 0.571i)15-s + (0.820 − 0.571i)16-s + (−0.954 + 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.260 + 0.965i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.260 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3456673214 - 0.2647881261i\)
\(L(\frac12)\) \(\approx\) \(-0.3456673214 - 0.2647881261i\)
\(L(1)\) \(\approx\) \(0.2628849916 - 0.5728090352i\)
\(L(1)\) \(\approx\) \(0.2628849916 - 0.5728090352i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.151 - 0.988i)T \)
3 \( 1 + (-0.0506 - 0.998i)T \)
5 \( 1 + (-0.528 - 0.848i)T \)
7 \( 1 + (-0.250 - 0.968i)T \)
11 \( 1 + (0.440 - 0.897i)T \)
13 \( 1 + (-0.440 + 0.897i)T \)
17 \( 1 + (-0.954 + 0.299i)T \)
19 \( 1 + (-0.820 - 0.571i)T \)
23 \( 1 + (0.0506 - 0.998i)T \)
29 \( 1 + (0.688 + 0.724i)T \)
31 \( 1 + (0.347 - 0.937i)T \)
37 \( 1 + (-0.612 + 0.790i)T \)
41 \( 1 + (0.979 - 0.201i)T \)
43 \( 1 + (0.954 - 0.299i)T \)
47 \( 1 + (-0.918 - 0.394i)T \)
53 \( 1 + (0.994 + 0.101i)T \)
59 \( 1 + (0.688 + 0.724i)T \)
61 \( 1 + (0.0506 + 0.998i)T \)
67 \( 1 + (-0.528 - 0.848i)T \)
71 \( 1 + (-0.954 - 0.299i)T \)
73 \( 1 + (-0.994 - 0.101i)T \)
79 \( 1 + (0.758 - 0.651i)T \)
83 \( 1 + (-0.994 - 0.101i)T \)
89 \( 1 + T \)
97 \( 1 + (0.954 - 0.299i)T \)
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

−25.36394738692760058486281500151, −24.75674693530368580048362911043, −23.229239930774837869067393671522, −22.7501675203347931942792259346, −22.140226973451878846452734797700, −21.26086777130357223543992814919, −19.74008977576703986680840153540, −19.226276406627661676312769204100, −17.75802631118354588490186517526, −17.57473629932830171114084441066, −16.015813887369855907482118140667, −15.57960451867627517900286762416, −14.886233991519594810689028654739, −14.28928415601378329822279899951, −12.776058851039197330617814033554, −11.71742519683597856312354115837, −10.50367784241272481339260085226, −9.734455194630940277452112029818, −8.83273804708040363832459733956, −7.85233400052886437068653932056, −6.70646786958429588248724247253, −5.8052568372263962174737826475, −4.742102725985429259841274938017, −3.792671868804881973508270737184, −2.55763856531526394163828040991, 0.29937332699587322705827354326, 1.31202307399726735749983585798, 2.56548568415313353932964752256, 3.94362849834293159090502056675, 4.703733898653244545201127869189, 6.31593154889208727908162953989, 7.381652352302916462433896621, 8.564416986321809795622941379800, 8.978644402721633387851124945318, 10.53515133132110605069968037202, 11.40196915817887163957331490965, 12.13285027573915775389610745087, 13.092010331243928925439817994021, 13.58373036400165839568788925328, 14.58751998235932154610281698020, 16.43118732225746303875206872467, 16.976389218193995623824762271212, 17.79115246077275251476896744474, 19.18217564510057622697918805472, 19.37332074138608243521866011217, 20.16211646571274736394235053778, 21.06858217181915164060639679600, 22.2125206780808569157385848253, 23.06262398921953787081792669621, 24.08915618928890815126494188376

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