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Properties

Label 1-1127-1127.372-r0-0-0
Degree $1$
Conductor $1127$
Sign $-0.592 - 0.805i$
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  + (−0.999 − 0.0407i)2-s + (−0.523 + 0.852i)3-s + (0.996 + 0.0815i)4-s + (−0.768 − 0.639i)5-s + (0.557 − 0.830i)6-s + (−0.992 − 0.122i)8-s + (−0.452 − 0.891i)9-s + (0.742 + 0.670i)10-s + (0.182 − 0.983i)11-s + (−0.591 + 0.806i)12-s + (−0.377 − 0.925i)13-s + (0.947 − 0.320i)15-s + (0.986 + 0.162i)16-s + (0.996 − 0.0815i)17-s + (0.415 + 0.909i)18-s + (−0.142 − 0.989i)19-s + ⋯
L(s)  = 1  + (−0.999 − 0.0407i)2-s + (−0.523 + 0.852i)3-s + (0.996 + 0.0815i)4-s + (−0.768 − 0.639i)5-s + (0.557 − 0.830i)6-s + (−0.992 − 0.122i)8-s + (−0.452 − 0.891i)9-s + (0.742 + 0.670i)10-s + (0.182 − 0.983i)11-s + (−0.591 + 0.806i)12-s + (−0.377 − 0.925i)13-s + (0.947 − 0.320i)15-s + (0.986 + 0.162i)16-s + (0.996 − 0.0815i)17-s + (0.415 + 0.909i)18-s + (−0.142 − 0.989i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $-0.592 - 0.805i$
Analytic conductor: \(5.23376\)
Root analytic conductor: \(5.23376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (372, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1127,\ (0:\ ),\ -0.592 - 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1748602365 - 0.3456677562i\)
\(L(\frac12)\) \(\approx\) \(0.1748602365 - 0.3456677562i\)
\(L(1)\) \(\approx\) \(0.4777150104 - 0.06903410786i\)
\(L(1)\) \(\approx\) \(0.4777150104 - 0.06903410786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.999 - 0.0407i)T \)
3 \( 1 + (-0.523 + 0.852i)T \)
5 \( 1 + (-0.768 - 0.639i)T \)
11 \( 1 + (0.182 - 0.983i)T \)
13 \( 1 + (-0.377 - 0.925i)T \)
17 \( 1 + (0.996 - 0.0815i)T \)
19 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + (0.996 - 0.0815i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (-0.979 - 0.202i)T \)
41 \( 1 + (-0.768 - 0.639i)T \)
43 \( 1 + (-0.992 + 0.122i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (-0.0611 - 0.998i)T \)
59 \( 1 + (0.742 + 0.670i)T \)
61 \( 1 + (0.339 - 0.940i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (-0.591 - 0.806i)T \)
73 \( 1 + (-0.862 - 0.505i)T \)
79 \( 1 + (-0.959 - 0.281i)T \)
83 \( 1 + (-0.452 - 0.891i)T \)
89 \( 1 + (0.101 - 0.994i)T \)
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

−21.58329723834109871733691488669, −20.55142788172254587567122194643, −19.72728052624442128205449186739, −19.038852306754636361499421036, −18.65605838964781275904805774533, −17.88088377976950285855144375417, −17.0178609035252635320342912495, −16.52531439539776173992371169468, −15.53105764415672683010867529773, −14.70342735571980798963991393957, −13.98180169512709065244190105120, −12.51395870327031423319212541983, −11.91436421288734740756085777102, −11.58880470633205390107023088252, −10.33955469421835869901732303544, −9.987449625455764638948577909857, −8.55838804649841579646193291359, −7.91721973709185847371246349742, −7.07071188584865938866030335211, −6.72015101075151809589597514502, −5.689969458994492928971249503518, −4.40954638250225073612081341117, −3.1075593260769753950539160414, −2.10101471619479637722262548421, −1.23956700799928603640448454634, 0.29810226435916304614856238791, 1.10447256162879051641015481832, 2.94776186403811671347311761344, 3.51424919587691503151412754073, 4.81566437811207835600563100240, 5.55903518486764016374302141290, 6.56096222265737321006797277487, 7.60329482290676871745271639226, 8.54262058621782532451097414114, 8.93470326018465860862155171830, 10.11547558152154735809310574990, 10.541925127770988323743913005596, 11.661201208051850347953988751061, 11.89132160659119637506186685093, 12.937497878855896551892288061531, 14.35208592429468806998220134141, 15.3129993179888329675556839649, 15.84937272839806946475617844420, 16.40672559879406291479573812043, 17.23260921184185783513705720313, 17.677702015122864819468133151777, 18.90272410958542967486780367702, 19.50716579631343893233935152735, 20.24824506356357548071179161032, 20.96886693316353820927848164374

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