↑

Properties

Label 1-373-373.22-r0-0-0
Degree $1$
Conductor $373$
Sign $0.00275 - 0.999i$
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  + (−0.528 − 0.848i)2-s + (−0.758 + 0.651i)3-s + (−0.440 + 0.897i)4-s + (0.0506 − 0.998i)5-s + (0.954 + 0.299i)6-s + (0.918 − 0.394i)7-s + (0.994 − 0.101i)8-s + (0.151 − 0.988i)9-s + (−0.874 + 0.485i)10-s + (0.994 + 0.101i)11-s + (−0.250 − 0.968i)12-s + (−0.994 − 0.101i)13-s + (−0.820 − 0.571i)14-s + (0.612 + 0.790i)15-s + (−0.612 − 0.790i)16-s + (−0.440 + 0.897i)17-s + ⋯
L(s)  = 1  + (−0.528 − 0.848i)2-s + (−0.758 + 0.651i)3-s + (−0.440 + 0.897i)4-s + (0.0506 − 0.998i)5-s + (0.954 + 0.299i)6-s + (0.918 − 0.394i)7-s + (0.994 − 0.101i)8-s + (0.151 − 0.988i)9-s + (−0.874 + 0.485i)10-s + (0.994 + 0.101i)11-s + (−0.250 − 0.968i)12-s + (−0.994 − 0.101i)13-s + (−0.820 − 0.571i)14-s + (0.612 + 0.790i)15-s + (−0.612 − 0.790i)16-s + (−0.440 + 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00275 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00275 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.00275 - 0.999i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.00275 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5582189213 - 0.5597606556i\)
\(L(\frac12)\) \(\approx\) \(0.5582189213 - 0.5597606556i\)
\(L(1)\) \(\approx\) \(0.6561685986 - 0.3013412405i\)
\(L(1)\) \(\approx\) \(0.6561685986 - 0.3013412405i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.528 - 0.848i)T \)
3 \( 1 + (-0.758 + 0.651i)T \)
5 \( 1 + (0.0506 - 0.998i)T \)
7 \( 1 + (0.918 - 0.394i)T \)
11 \( 1 + (0.994 + 0.101i)T \)
13 \( 1 + (-0.994 - 0.101i)T \)
17 \( 1 + (-0.440 + 0.897i)T \)
19 \( 1 + (0.612 - 0.790i)T \)
23 \( 1 + (0.758 + 0.651i)T \)
29 \( 1 + (0.347 - 0.937i)T \)
31 \( 1 + (-0.250 + 0.968i)T \)
37 \( 1 + (0.979 - 0.201i)T \)
41 \( 1 + (-0.954 - 0.299i)T \)
43 \( 1 + (0.440 - 0.897i)T \)
47 \( 1 + (-0.820 + 0.571i)T \)
53 \( 1 + (-0.151 - 0.988i)T \)
59 \( 1 + (0.347 - 0.937i)T \)
61 \( 1 + (0.758 - 0.651i)T \)
67 \( 1 + (0.0506 - 0.998i)T \)
71 \( 1 + (-0.440 - 0.897i)T \)
73 \( 1 + (0.151 + 0.988i)T \)
79 \( 1 + (0.874 - 0.485i)T \)
83 \( 1 + (0.151 + 0.988i)T \)
89 \( 1 + T \)
97 \( 1 + (0.440 - 0.897i)T \)
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

−24.78559287360453093968382983894, −24.21379196024390057938583650727, −23.15283135831058893398587258966, −22.40920649840016039080198756823, −21.84354743141154382731462852248, −20.13010474793838235284222941619, −19.09379770889282477026678443584, −18.41330127199236309361791659122, −17.8429741085983862506219101755, −17.02160590531269212239271713327, −16.213626175624739809533251920710, −14.79140744489758384009790065083, −14.4995441371976015783828374028, −13.42849263133169751142145930916, −11.90613404258506774605756420024, −11.30323020338922723800966025716, −10.29061290807674977968222965293, −9.215640438808072461817127676449, −7.93936212655111345226884082711, −7.17910239513392237154006916982, −6.4675348532811258326732410342, −5.46149801521174037741259501799, −4.52305279971476070246475211271, −2.44517599106460250720245883279, −1.235382446119010481304329611957, 0.7605918485376511195701634572, 1.819096186445791002408016384390, 3.64685620003665011557658583696, 4.57648598622256635711572622673, 5.17216041590295004502405823973, 6.875111029321902827230014853125, 8.11337358733514581913017538857, 9.13995686152925491447474471882, 9.76585116501972245499784931842, 10.88824520250504343701764097690, 11.641538740572736762033874887584, 12.298413207663111877152909664385, 13.336559413905792568667953594571, 14.58628630404739960320502117742, 15.763523207245980536741557751593, 16.939683771062296193032996359048, 17.287250843756175779753401096536, 17.83142743511242696507860042830, 19.44947831465236502771530056889, 20.050847344377462926152774363183, 20.9209872134928733938100360904, 21.64441214395301889597774561267, 22.27589979371181628872079919214, 23.468863200695898836897622376233, 24.29877652513633832950302475560

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