↑

Properties

Label 1-373-373.18-r1-0-0
Degree $1$
Conductor $373$
Sign $0.774 - 0.632i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (−0.201 + 0.979i)2-s + (0.440 + 0.897i)3-s + (−0.918 − 0.394i)4-s + (−0.299 − 0.954i)5-s + (−0.968 + 0.250i)6-s + (0.758 − 0.651i)7-s + (0.571 − 0.820i)8-s + (−0.612 + 0.790i)9-s + (0.994 − 0.101i)10-s + (0.571 + 0.820i)11-s + (−0.0506 − 0.998i)12-s + (−0.820 + 0.571i)13-s + (0.485 + 0.874i)14-s + (0.724 − 0.688i)15-s + (0.688 + 0.724i)16-s + (−0.918 − 0.394i)17-s + ⋯
L(s)  = 1  + (−0.201 + 0.979i)2-s + (0.440 + 0.897i)3-s + (−0.918 − 0.394i)4-s + (−0.299 − 0.954i)5-s + (−0.968 + 0.250i)6-s + (0.758 − 0.651i)7-s + (0.571 − 0.820i)8-s + (−0.612 + 0.790i)9-s + (0.994 − 0.101i)10-s + (0.571 + 0.820i)11-s + (−0.0506 − 0.998i)12-s + (−0.820 + 0.571i)13-s + (0.485 + 0.874i)14-s + (0.724 − 0.688i)15-s + (0.688 + 0.724i)16-s + (−0.918 − 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.774 - 0.632i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.774 - 0.632i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8164753271 - 0.2908500885i\)
\(L(\frac12)\) \(\approx\) \(0.8164753271 - 0.2908500885i\)
\(L(1)\) \(\approx\) \(0.7969366668 + 0.3602299424i\)
\(L(1)\) \(\approx\) \(0.7969366668 + 0.3602299424i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.201 + 0.979i)T \)
3 \( 1 + (0.440 + 0.897i)T \)
5 \( 1 + (-0.299 - 0.954i)T \)
7 \( 1 + (0.758 - 0.651i)T \)
11 \( 1 + (0.571 + 0.820i)T \)
13 \( 1 + (-0.820 + 0.571i)T \)
17 \( 1 + (-0.918 - 0.394i)T \)
19 \( 1 + (-0.724 - 0.688i)T \)
23 \( 1 + (0.897 + 0.440i)T \)
29 \( 1 + (0.528 + 0.848i)T \)
31 \( 1 + (0.0506 - 0.998i)T \)
37 \( 1 + (-0.347 - 0.937i)T \)
41 \( 1 + (-0.250 - 0.968i)T \)
43 \( 1 + (0.394 - 0.918i)T \)
47 \( 1 + (-0.485 + 0.874i)T \)
53 \( 1 + (0.790 - 0.612i)T \)
59 \( 1 + (-0.528 - 0.848i)T \)
61 \( 1 + (0.897 - 0.440i)T \)
67 \( 1 + (-0.299 - 0.954i)T \)
71 \( 1 + (-0.918 + 0.394i)T \)
73 \( 1 + (-0.612 - 0.790i)T \)
79 \( 1 + (-0.101 - 0.994i)T \)
83 \( 1 + (-0.612 - 0.790i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.394 + 0.918i)T \)
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

−24.59158421577593436023202919197, −23.488459998879834797299800628663, −22.62222596440153619098220397681, −21.77003807502788166041504083193, −20.98436642622703025608558220370, −19.72524590501015775019624865286, −19.32004084253597255154220627352, −18.50400126785656184740721170842, −17.82638870663735401271720645931, −17.01970184464938962936484410874, −15.09287298788763074657219039641, −14.55737346192098097642743009331, −13.6575096996201407344842473522, −12.61612728197910349734726056161, −11.77085458952496017889299162221, −11.13042837660644921055975446412, −10.06252478203603459613506060426, −8.62121701379418528145438037883, −8.28627499811414254369320041440, −7.0528798043450316458806376806, −5.900139656973239067955590404383, −4.37054180196689644081112578363, −3.06288161794657765295287302533, −2.43894514954222995430142039665, −1.26483010442387452165028557462, 0.25540149361987090264925023698, 1.88774532471906469253574138609, 3.993559022052853730509597366704, 4.60910757938337922654952886556, 5.17025172040354909164577953829, 6.92093004031248898886518822690, 7.701937119507629666327897927384, 8.91288950699298686070816427507, 9.191409454414857995296019540355, 10.386911969465094666404206972650, 11.50843440594191120358395376995, 12.895131035809068900913152262498, 13.87847873398420104805733789460, 14.705666065932475407171230737268, 15.40518823966344108927964998966, 16.28846438386045306689554567651, 17.27731112377538228661304541100, 17.426628853489616234680738219515, 19.25528546019377517076266583844, 19.891773331544683516314105400834, 20.74154555528923929378676602605, 21.73379538006080202037917905029, 22.66438157899860757969416276141, 23.660208158556587095848070930746, 24.38332869945184935628009953612

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