↑

Properties

Label 1-407-407.200-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.656 + 0.754i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (−0.970 + 0.241i)2-s + (0.374 + 0.927i)3-s + (0.882 − 0.469i)4-s + (−0.275 − 0.961i)5-s + (−0.587 − 0.809i)6-s + (−0.0348 + 0.999i)7-s + (−0.743 + 0.669i)8-s + (−0.719 + 0.694i)9-s + (0.5 + 0.866i)10-s + (0.766 + 0.642i)12-s + (0.898 + 0.438i)13-s + (−0.207 − 0.978i)14-s + (0.788 − 0.615i)15-s + (0.559 − 0.829i)16-s + (−0.898 + 0.438i)17-s + (0.529 − 0.848i)18-s + ⋯
L(s)  = 1  + (−0.970 + 0.241i)2-s + (0.374 + 0.927i)3-s + (0.882 − 0.469i)4-s + (−0.275 − 0.961i)5-s + (−0.587 − 0.809i)6-s + (−0.0348 + 0.999i)7-s + (−0.743 + 0.669i)8-s + (−0.719 + 0.694i)9-s + (0.5 + 0.866i)10-s + (0.766 + 0.642i)12-s + (0.898 + 0.438i)13-s + (−0.207 − 0.978i)14-s + (0.788 − 0.615i)15-s + (0.559 − 0.829i)16-s + (−0.898 + 0.438i)17-s + (0.529 − 0.848i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.656 + 0.754i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.656 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3001155847 + 0.6593423910i\)
\(L(\frac12)\) \(\approx\) \(0.3001155847 + 0.6593423910i\)
\(L(1)\) \(\approx\) \(0.6187112771 + 0.3460352902i\)
\(L(1)\) \(\approx\) \(0.6187112771 + 0.3460352902i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.970 + 0.241i)T \)
3 \( 1 + (0.374 + 0.927i)T \)
5 \( 1 + (-0.275 - 0.961i)T \)
7 \( 1 + (-0.0348 + 0.999i)T \)
13 \( 1 + (0.898 + 0.438i)T \)
17 \( 1 + (-0.898 + 0.438i)T \)
19 \( 1 + (0.927 - 0.374i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.207 + 0.978i)T \)
31 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.990 + 0.139i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (0.961 + 0.275i)T \)
59 \( 1 + (-0.788 + 0.615i)T \)
61 \( 1 + (-0.694 + 0.719i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (-0.241 + 0.970i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.829 + 0.559i)T \)
83 \( 1 + (-0.438 - 0.898i)T \)
89 \( 1 + (-0.342 - 0.939i)T \)
97 \( 1 + (0.994 + 0.104i)T \)
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

−24.23923057094518838755003374168, −23.15437052811732714823445971020, −22.487038113606085079697903192182, −21.00508295612376884019247823123, −20.15276045616442314629521990318, −19.663099895338281238859649542780, −18.662500675837470177632126577190, −18.064679704006858862604103078701, −17.451034243415030324711473523788, −16.19409997623329594839169043458, −15.3388974682920615253849020572, −14.10818618571146827785709818065, −13.450056332634746397049359195953, −12.20353305017272741924271044526, −11.3179773176369315155742281452, −10.57684823130008185706895980632, −9.57903556375640392747069762465, −8.30346600811080053708856829899, −7.670058646612553401526081739014, −6.84064671449303849962819348103, −6.14648256201931984647073115648, −3.83333727169759749431949390669, −3.00356193493334369709816305505, −1.87948872979266061601466021319, −0.55885200842897778825025172316, 1.52929961801379970935737656370, 2.76704127994436869580131522646, 4.1350498042665568906561017456, 5.32437728893679671047525036302, 6.14550620178710307088216012617, 7.75564665675735588136881148413, 8.6323114984664586736198946153, 9.09600405426469992986552546906, 9.8837391800019358897958283159, 11.21054112878637187106079545620, 11.726327815903049096853341177752, 13.109849063492920295770373232714, 14.35800724478377625378420862168, 15.4899280109848356878947889593, 15.84348915594974741133742499215, 16.54312371173771422473253114017, 17.61132780822370499145340840012, 18.50543058836609324269348053152, 19.65716326174262041088720619641, 20.086363167764176319380363785646, 21.08558490745747872479826431576, 21.67387850045701345305819550820, 22.94528196647784985686386833044, 24.277957522209983959304005954951, 24.62051076069590404815435219531

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