↑

Properties

Label 1-403-403.197-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.634 + 0.772i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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  + (−0.994 + 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 − 0.207i)4-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + (−0.743 + 0.669i)7-s + (−0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (−0.913 − 0.406i)10-s + (0.207 + 0.978i)11-s + (0.913 + 0.406i)12-s + (0.669 − 0.743i)14-s + (0.406 + 0.913i)15-s + (0.913 − 0.406i)16-s + (−0.978 − 0.207i)17-s + (−0.406 − 0.913i)18-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 − 0.207i)4-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + (−0.743 + 0.669i)7-s + (−0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (−0.913 − 0.406i)10-s + (0.207 + 0.978i)11-s + (0.913 + 0.406i)12-s + (0.669 − 0.743i)14-s + (0.406 + 0.913i)15-s + (0.913 − 0.406i)16-s + (−0.978 − 0.207i)17-s + (−0.406 − 0.913i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.634 + 0.772i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.634 + 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4426404947 + 0.9359375640i\)
\(L(\frac12)\) \(\approx\) \(0.4426404947 + 0.9359375640i\)
\(L(1)\) \(\approx\) \(0.7715916651 + 0.4834334129i\)
\(L(1)\) \(\approx\) \(0.7715916651 + 0.4834334129i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.994 + 0.104i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.743 + 0.669i)T \)
11 \( 1 + (0.207 + 0.978i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (-0.994 + 0.104i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.406 - 0.913i)T \)
43 \( 1 + (-0.104 - 0.994i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (0.406 + 0.913i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (0.743 - 0.669i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (0.994 - 0.104i)T \)
89 \( 1 + (0.743 - 0.669i)T \)
97 \( 1 + (0.207 + 0.978i)T \)
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

−24.39111741861914160453238702903, −23.61057895770260653073718195736, −21.976432321242246931611278300782, −21.16764344219447563225033378590, −20.28242484165606220786145175121, −19.62359810638522151036210637732, −18.96781554884182716334636735624, −17.93266152087954908752455625134, −17.19501075015517749127266692743, −16.37811427066347803797869115515, −15.4024278853180956762008344705, −14.13092881008508343272189527016, −13.30511605046235390486000421674, −12.642976032783496462512512438995, −11.35795524337615744431514844116, −10.18321073494232733573588444681, −9.46394811883028231348371060039, −8.63000171931866961908002997244, −7.892885757323603436816731489004, −6.483304459430025478081779434220, −6.2342166925717713710449235740, −4.063906665823757726315636008386, −2.846600537305243176095540907103, −1.88696134921801230653504264004, −0.70891705715969464466756451411, 2.13559970865784734594173224382, 2.36461309166729847387558210185, 3.81622369603057743268762551814, 5.42441972140760646981403540822, 6.535411311831866790225350990822, 7.33010929758845578736876409486, 8.81788119618362137682795398740, 9.1278353986345059745603741120, 10.18653890872368749848145783381, 10.599151734980742268859113708612, 12.10938289286179899934366213346, 13.17993180219078368539564047469, 14.395916267627608266227490956282, 15.11451434015354641362201617069, 15.8352753687732429913729097957, 16.79683665729780568952399259068, 17.8290675436702924215929290358, 18.52453892948043851530910900499, 19.49634508510829532945334268561, 20.12452567241432846483958422180, 21.10279391072895374914129287916, 21.87056916002886580295640503763, 22.665082240404567020147298386145, 24.24960249340003911720468046450, 25.22935812083523036248043488565

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