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Properties

Label 1-1127-1127.40-r0-0-0
Degree $1$
Conductor $1127$
Sign $-0.999 + 0.0288i$
Analytic cond. $5.23376$
Root an. cond. $5.23376$
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  + (0.704 − 0.709i)2-s + (−0.644 − 0.764i)3-s + (−0.00679 − 0.999i)4-s + (0.314 + 0.949i)5-s + (−0.996 − 0.0815i)6-s + (−0.714 − 0.699i)8-s + (−0.169 + 0.985i)9-s + (0.894 + 0.446i)10-s + (0.115 − 0.993i)11-s + (−0.760 + 0.649i)12-s + (−0.986 − 0.162i)13-s + (0.523 − 0.852i)15-s + (−0.999 + 0.0135i)16-s + (0.869 − 0.494i)17-s + (0.580 + 0.814i)18-s + (0.928 − 0.371i)19-s + ⋯
L(s)  = 1  + (0.704 − 0.709i)2-s + (−0.644 − 0.764i)3-s + (−0.00679 − 0.999i)4-s + (0.314 + 0.949i)5-s + (−0.996 − 0.0815i)6-s + (−0.714 − 0.699i)8-s + (−0.169 + 0.985i)9-s + (0.894 + 0.446i)10-s + (0.115 − 0.993i)11-s + (−0.760 + 0.649i)12-s + (−0.986 − 0.162i)13-s + (0.523 − 0.852i)15-s + (−0.999 + 0.0135i)16-s + (0.869 − 0.494i)17-s + (0.580 + 0.814i)18-s + (0.928 − 0.371i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $-0.999 + 0.0288i$
Analytic conductor: \(5.23376\)
Root analytic conductor: \(5.23376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1127,\ (0:\ ),\ -0.999 + 0.0288i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01893904772 - 1.313945199i\)
\(L(\frac12)\) \(\approx\) \(0.01893904772 - 1.313945199i\)
\(L(1)\) \(\approx\) \(0.8384143434 - 0.7799998930i\)
\(L(1)\) \(\approx\) \(0.8384143434 - 0.7799998930i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.704 - 0.709i)T \)
3 \( 1 + (-0.644 - 0.764i)T \)
5 \( 1 + (0.314 + 0.949i)T \)
11 \( 1 + (0.115 - 0.993i)T \)
13 \( 1 + (-0.986 - 0.162i)T \)
17 \( 1 + (0.869 - 0.494i)T \)
19 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (-0.862 - 0.505i)T \)
31 \( 1 + (-0.0475 - 0.998i)T \)
37 \( 1 + (0.275 + 0.961i)T \)
41 \( 1 + (0.979 + 0.202i)T \)
43 \( 1 + (0.714 - 0.699i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (-0.990 - 0.135i)T \)
59 \( 1 + (-0.894 - 0.446i)T \)
61 \( 1 + (-0.912 + 0.409i)T \)
67 \( 1 + (-0.981 + 0.189i)T \)
71 \( 1 + (0.182 - 0.983i)T \)
73 \( 1 + (-0.976 - 0.215i)T \)
79 \( 1 + (-0.723 - 0.690i)T \)
83 \( 1 + (-0.768 - 0.639i)T \)
89 \( 1 + (0.390 + 0.920i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−21.72965968313200238927304494835, −21.20812304361652427051947366134, −20.49075509931242362299298543894, −19.78659010572368773974969522143, −18.21303009537826235893048270356, −17.40865768242044698359783452976, −17.037550734805966796474730700349, −16.18319584866332427289078258473, −15.75531189334350333518497179595, −14.55433076526732914136655333954, −14.36171956995086508190757055941, −12.88268800851563424588953419173, −12.39342478071435866771181279196, −11.88108842659354147297351388964, −10.67803061582453795196977644037, −9.51339663639878549992029668156, −9.24865229343128981610307328264, −7.894365687897039047203829766525, −7.18091211578430839085446344401, −5.98470309416309961450587264335, −5.41756652468665593615380142271, −4.671845372199521589385644173319, −4.06994081481135999074233701759, −2.92675082334651247421663116469, −1.45942616667726985419669960672, 0.45704917275245647703950023665, 1.65334331942634297088833839721, 2.67787987229619703757073760671, 3.25490787838083375943916817940, 4.63136865253806746296742062664, 5.67739957894229393366173964653, 6.00171667817722414876188604426, 7.11927033696553710907866387971, 7.734373572546369751605255698249, 9.34119392200137326287612262543, 10.08428290679860961413203912463, 10.95158240860613328611445488452, 11.56404690335078667444786429041, 12.12780991924595602797047364786, 13.19776081646813208478794273346, 13.76247816630835533626638085987, 14.40206190541363517167491051941, 15.25846595233371204224227353351, 16.35508396600599579422882272494, 17.23690693100267371164756374498, 18.16261436804328858567674524846, 18.798623738084744857301016821852, 19.21044450488506996548563462923, 20.14526340928416565719941830384, 21.149767305399594184025436731719

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