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Properties

Label 1-967-967.677-r1-0-0
Degree $1$
Conductor $967$
Sign $0.606 - 0.795i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (0.389 − 0.921i)2-s + (0.999 − 0.0195i)3-s + (−0.696 − 0.717i)4-s + (−0.763 − 0.646i)5-s + (0.371 − 0.928i)6-s + (0.998 − 0.0585i)7-s + (−0.932 + 0.362i)8-s + (0.999 − 0.0390i)9-s + (−0.892 + 0.451i)10-s + (−0.184 + 0.982i)11-s + (−0.710 − 0.703i)12-s + (0.184 + 0.982i)13-s + (0.334 − 0.942i)14-s + (−0.775 − 0.631i)15-s + (−0.0292 + 0.999i)16-s + (−0.442 − 0.896i)17-s + ⋯
L(s)  = 1  + (0.389 − 0.921i)2-s + (0.999 − 0.0195i)3-s + (−0.696 − 0.717i)4-s + (−0.763 − 0.646i)5-s + (0.371 − 0.928i)6-s + (0.998 − 0.0585i)7-s + (−0.932 + 0.362i)8-s + (0.999 − 0.0390i)9-s + (−0.892 + 0.451i)10-s + (−0.184 + 0.982i)11-s + (−0.710 − 0.703i)12-s + (0.184 + 0.982i)13-s + (0.334 − 0.942i)14-s + (−0.775 − 0.631i)15-s + (−0.0292 + 0.999i)16-s + (−0.442 − 0.896i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.606 - 0.795i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.606 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.146654815 - 1.557799634i\)
\(L(\frac12)\) \(\approx\) \(3.146654815 - 1.557799634i\)
\(L(1)\) \(\approx\) \(1.508848714 - 0.7876721283i\)
\(L(1)\) \(\approx\) \(1.508848714 - 0.7876721283i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.389 - 0.921i)T \)
3 \( 1 + (0.999 - 0.0195i)T \)
5 \( 1 + (-0.763 - 0.646i)T \)
7 \( 1 + (0.998 - 0.0585i)T \)
11 \( 1 + (-0.184 + 0.982i)T \)
13 \( 1 + (0.184 + 0.982i)T \)
17 \( 1 + (-0.442 - 0.896i)T \)
19 \( 1 + (0.638 + 0.769i)T \)
23 \( 1 + (-0.316 - 0.948i)T \)
29 \( 1 + (-0.592 - 0.805i)T \)
31 \( 1 + (0.560 + 0.828i)T \)
37 \( 1 + (0.883 + 0.468i)T \)
41 \( 1 + (0.0682 + 0.997i)T \)
43 \( 1 + (-0.787 + 0.615i)T \)
47 \( 1 + (-0.00975 + 0.999i)T \)
53 \( 1 + (0.854 - 0.519i)T \)
59 \( 1 + (0.938 - 0.344i)T \)
61 \( 1 + (-0.0682 - 0.997i)T \)
67 \( 1 + (-0.560 + 0.828i)T \)
71 \( 1 + (0.389 + 0.921i)T \)
73 \( 1 + (0.854 + 0.519i)T \)
79 \( 1 + (0.995 + 0.0974i)T \)
83 \( 1 + (0.987 + 0.155i)T \)
89 \( 1 + (-0.560 + 0.828i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−21.80537034554396500024025482203, −21.029161250408462753249975365128, −20.06693712841231996309906906360, −19.32602030022911179135407614496, −18.27326988253378922804542194503, −17.976640159591607192491910641392, −16.74994208379226672517338788389, −15.73569075620607284072902973618, −15.17409624735160780062801426777, −14.80783469662726313616038562086, −13.72335380021038700149180780279, −13.396106521762966213682062768716, −12.2026537276858747899421534736, −11.24308890586361838082013236202, −10.374980240306925332948512055175, −9.02296666605736317182824884988, −8.36330110138169057692390469764, −7.750063194916907807877060412054, −7.18367239182742210618149447875, −5.96487386058898984544711135008, −5.02144231967807148751926320446, −3.89522572158331070242499467509, −3.407496878093528843765120595184, −2.355696346692613858540162960351, −0.676296439335741261887537715007, 0.98350221821918631121466232656, 1.806590092366538481669996417088, 2.66900258213723849233705143559, 3.90975597020611571150662199213, 4.48274232887105529302212413595, 5.05378367326383916975856615462, 6.72968836473996280368766929659, 7.85326474722005473042784970054, 8.41846488279914721912845357962, 9.37505769540672438906182938049, 9.96678109987901271999232435641, 11.234690934971055548668394823263, 11.83252068361128949917704282092, 12.593031243673884478290796550415, 13.445485218714755030827658334471, 14.24445106561373985349265681039, 14.82299609581734840920307828882, 15.63211573568266506783370423579, 16.58556677845881413564599907157, 17.96700027002954227050785425703, 18.454124240259293762857104078333, 19.34801217929738372473491767628, 20.107137240494468016810373076023, 20.71670734625494173591017985937, 20.92456061878577310324387565603

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