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Properties

Label 1-547-547.375-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.0523 - 0.998i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.120 − 0.992i)2-s + 3-s + (−0.970 − 0.239i)4-s + (−0.354 − 0.935i)5-s + (0.120 − 0.992i)6-s + (0.120 + 0.992i)7-s + (−0.354 + 0.935i)8-s + 9-s + (−0.970 + 0.239i)10-s + (0.568 − 0.822i)11-s + (−0.970 − 0.239i)12-s + 13-s + 14-s + (−0.354 − 0.935i)15-s + (0.885 + 0.464i)16-s + (0.568 + 0.822i)17-s + ⋯
L(s)  = 1  + (0.120 − 0.992i)2-s + 3-s + (−0.970 − 0.239i)4-s + (−0.354 − 0.935i)5-s + (0.120 − 0.992i)6-s + (0.120 + 0.992i)7-s + (−0.354 + 0.935i)8-s + 9-s + (−0.970 + 0.239i)10-s + (0.568 − 0.822i)11-s + (−0.970 − 0.239i)12-s + 13-s + 14-s + (−0.354 − 0.935i)15-s + (0.885 + 0.464i)16-s + (0.568 + 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0523 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0523 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.0523 - 0.998i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (375, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.0523 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.359100926 - 1.432174932i\)
\(L(\frac12)\) \(\approx\) \(1.359100926 - 1.432174932i\)
\(L(1)\) \(\approx\) \(1.240692436 - 0.8066448089i\)
\(L(1)\) \(\approx\) \(1.240692436 - 0.8066448089i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.120 - 0.992i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.354 - 0.935i)T \)
7 \( 1 + (0.120 + 0.992i)T \)
11 \( 1 + (0.568 - 0.822i)T \)
13 \( 1 + T \)
17 \( 1 + (0.568 + 0.822i)T \)
19 \( 1 + (0.885 - 0.464i)T \)
23 \( 1 + (-0.748 + 0.663i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (0.885 + 0.464i)T \)
37 \( 1 + (-0.748 - 0.663i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (-0.748 - 0.663i)T \)
53 \( 1 + (-0.354 - 0.935i)T \)
59 \( 1 + (0.885 + 0.464i)T \)
61 \( 1 + (0.120 - 0.992i)T \)
67 \( 1 + (-0.970 - 0.239i)T \)
71 \( 1 + (0.885 - 0.464i)T \)
73 \( 1 + (-0.354 + 0.935i)T \)
79 \( 1 + (-0.970 - 0.239i)T \)
83 \( 1 + (-0.354 - 0.935i)T \)
89 \( 1 + (0.120 + 0.992i)T \)
97 \( 1 + (0.568 + 0.822i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.65078714368225226506774888209, −22.7814097557910397006216475908, −22.352474378650421814699075652456, −20.952909792165390502926686848700, −20.31164867599791328958228332718, −19.28983123320473867581808990146, −18.39369693131931373279878360792, −17.89494658718861972864125840206, −16.58631200462178219783392545632, −15.86456303194901229129833311805, −15.006623382956210256156266501685, −14.137041392498519488992921861062, −13.96166211958350972442350338412, −12.8282467748074418103525511991, −11.632039939514568562615987348960, −10.21343065853114184374567314969, −9.6911919991282940645008013014, −8.46821774625952760974959506816, −7.59114808867963821538759221909, −7.12807636553560257074666586979, −6.20692966699342671162016881471, −4.60175223101823401392915073871, −3.82550786549680940200330641536, −3.08849316954073376675329120606, −1.33216625411158011283923992784, 1.147050334586206579374570164804, 1.97841733558472029521670386296, 3.36128326812525458006283784771, 3.82621135608252796267019357875, 5.090558097908340156693183936202, 6.01976185141326654458181534751, 7.937102224584794713987869644195, 8.55200960632488138827451833809, 9.12591161698605080353681809745, 9.97296065965184246055613414158, 11.36150367615267370148841921939, 11.97240588627994015217870724377, 12.92405584583526342156270336948, 13.58913641898501807239530298742, 14.43227377322926108347932082400, 15.46836488440447955428051559396, 16.19888277972857143038881751075, 17.547336517658942464987636075635, 18.50300743744311246869884447183, 19.283145693849158105385420768105, 19.74204876854949138628709450422, 20.74957240670267310629765237477, 21.27335868103250727408797325356, 21.91819604319788046918245522772, 23.10319651861198149313155406864

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