Properties

Label 1-43-43.38-r0-0-0
Degree $1$
Conductor $43$
Sign $0.739 + 0.673i$
Analytic cond. $0.199691$
Root an. cond. $0.199691$
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  + (−0.900 − 0.433i)2-s + (0.826 + 0.563i)3-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (−0.222 − 0.974i)8-s + (0.365 + 0.930i)9-s + (0.955 − 0.294i)10-s + (0.623 − 0.781i)11-s + (0.0747 + 0.997i)12-s + (0.955 + 0.294i)13-s + (0.826 − 0.563i)14-s + (−0.988 + 0.149i)15-s + (−0.222 + 0.974i)16-s + (−0.733 − 0.680i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.826 + 0.563i)3-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (−0.222 − 0.974i)8-s + (0.365 + 0.930i)9-s + (0.955 − 0.294i)10-s + (0.623 − 0.781i)11-s + (0.0747 + 0.997i)12-s + (0.955 + 0.294i)13-s + (0.826 − 0.563i)14-s + (−0.988 + 0.149i)15-s + (−0.222 + 0.974i)16-s + (−0.733 − 0.680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $0.739 + 0.673i$
Analytic conductor: \(0.199691\)
Root analytic conductor: \(0.199691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (0:\ ),\ 0.739 + 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6019071077 + 0.2331826758i\)
\(L(\frac12)\) \(\approx\) \(0.6019071077 + 0.2331826758i\)
\(L(1)\) \(\approx\) \(0.7655876424 + 0.1524738964i\)
\(L(1)\) \(\approx\) \(0.7655876424 + 0.1524738964i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T \)
3 \( 1 + (0.826 + 0.563i)T \)
5 \( 1 + (-0.733 + 0.680i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (0.955 + 0.294i)T \)
17 \( 1 + (-0.733 - 0.680i)T \)
19 \( 1 + (0.365 - 0.930i)T \)
23 \( 1 + (-0.988 - 0.149i)T \)
29 \( 1 + (0.826 - 0.563i)T \)
31 \( 1 + (0.0747 + 0.997i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (0.955 - 0.294i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (0.0747 - 0.997i)T \)
67 \( 1 + (0.365 - 0.930i)T \)
71 \( 1 + (-0.988 + 0.149i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.826 + 0.563i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−35.1690727568550672215671994553, −33.18677145222468076049730778134, −32.41809022754650976907837386444, −30.95138702578316208062848389136, −29.77228248258738255574269863030, −28.44636938277335740335975964302, −27.26587099189980965633680147453, −26.14911096093102759156593761533, −25.20658331988146734434412709802, −24.04875150425056773768840632621, −23.15053721534032075555175278308, −20.342772572598819944917162990282, −20.0717592747222592800325769888, −18.86532694366812845011164364354, −17.48635971980224012807316369442, −16.16871227611918924652338651453, −15.02485293585769421073382240828, −13.514858015302777328335759930019, −12.00092429281010479717529408880, −10.11152382814417587795531986584, −8.73514286029779617758797687686, −7.735936133521329107220922299369, −6.501738459213889005682048455025, −3.87772807411396579082970139048, −1.39909752181523223221470019462, 2.62523286655952035075974842607, 3.75816814825614387024191236852, 6.71548433181936992543005051613, 8.37472597309638274344771855324, 9.2169955810993646119576084162, 10.75035105593948037856763672479, 11.86804227566457767912495233933, 13.80718609333910882299293011705, 15.590017529971358466577824438872, 16.07158905360950267642983343013, 18.19913069436999997823070551365, 19.20630538006612677012928681673, 19.96677266510448469736257265474, 21.474911602967153480170094645, 22.34057302221512755772140568886, 24.56537773410241680145662937813, 25.75247125794290714376666475112, 26.60229788367684022953436265453, 27.53114542744559122737850235267, 28.639505248926069723416029826195, 30.28334439011530606059570457246, 31.03112053935256928899324799761, 32.26727118231562409588748676744, 33.873033918830098190314092897968, 35.02188403538532714266030818319

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