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Properties

Label 1-629-629.193-r0-0-0
Degree $1$
Conductor $629$
Sign $-0.392 + 0.919i$
Analytic cond. $2.92106$
Root an. cond. $2.92106$
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  + (0.258 + 0.965i)2-s + (0.793 + 0.608i)3-s + (−0.866 + 0.5i)4-s + (−0.793 − 0.608i)5-s + (−0.382 + 0.923i)6-s + (0.991 − 0.130i)7-s + (−0.707 − 0.707i)8-s + (0.258 + 0.965i)9-s + (0.382 − 0.923i)10-s + (0.923 + 0.382i)11-s + (−0.991 − 0.130i)12-s + (−0.5 − 0.866i)13-s + (0.382 + 0.923i)14-s + (−0.258 − 0.965i)15-s + (0.5 − 0.866i)16-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (0.793 + 0.608i)3-s + (−0.866 + 0.5i)4-s + (−0.793 − 0.608i)5-s + (−0.382 + 0.923i)6-s + (0.991 − 0.130i)7-s + (−0.707 − 0.707i)8-s + (0.258 + 0.965i)9-s + (0.382 − 0.923i)10-s + (0.923 + 0.382i)11-s + (−0.991 − 0.130i)12-s + (−0.5 − 0.866i)13-s + (0.382 + 0.923i)14-s + (−0.258 − 0.965i)15-s + (0.5 − 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 629 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(629\)    =    \(17 \cdot 37\)
Sign: $-0.392 + 0.919i$
Analytic conductor: \(2.92106\)
Root analytic conductor: \(2.92106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{629} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 629,\ (0:\ ),\ -0.392 + 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053640618 + 1.595875064i\)
\(L(\frac12)\) \(\approx\) \(1.053640618 + 1.595875064i\)
\(L(1)\) \(\approx\) \(1.126217373 + 0.8849245332i\)
\(L(1)\) \(\approx\) \(1.126217373 + 0.8849245332i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (0.793 + 0.608i)T \)
5 \( 1 + (-0.793 - 0.608i)T \)
7 \( 1 + (0.991 - 0.130i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.965 + 0.258i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (0.923 - 0.382i)T \)
31 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.991 + 0.130i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (0.965 - 0.258i)T \)
61 \( 1 + (0.130 + 0.991i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.793 - 0.608i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (-0.991 + 0.130i)T \)
83 \( 1 + (0.965 - 0.258i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.923 + 0.382i)T \)
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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

−22.44471487887170815254879781676, −21.91985721937094542709812928123, −20.82646578480722889261700613923, −20.21066947067961272976056467410, −19.377265889893969114499462049234, −18.8678408777794501492691421526, −18.14176517462718648131086969695, −17.26973892459162414521379638543, −15.7498780842241567720781648391, −14.56664051142071445055318928951, −14.40446877967679591107589830592, −13.62802399329480079581553966128, −12.25866379307160025263659179380, −11.82870291004881490475798428205, −11.14201680628015365096295150208, −9.9419342415605976993253639739, −8.92543472594255468088248805811, −8.25436307123541634053705990455, −7.24895326483697875339274408847, −6.2375439305634970574060944332, −4.676111861768818850463653603055, −3.939043320053738918017852210904, −2.9462215595022756078509209640, −2.065083144054354194109581889787, −0.97188145425126622949587897762, 1.31541958731280232241223268245, 3.11854743348796739022148157030, 4.01761081389211146888708810906, 4.76281166413164713134184212293, 5.43625451079050715994670025971, 7.09574045339260340230319853200, 7.8514912778170069855384184737, 8.37695870951681011525836288963, 9.27995102863277757885670379916, 10.16162406007496806025068625841, 11.614775684997285212305133957574, 12.33129589978452073955789485710, 13.51253819458314655969406959253, 14.24586361710010579592486053387, 14.98621483894388002263656231794, 15.58077520884376119147628128196, 16.3574441344007840717563668220, 17.269215284514884327897908438902, 17.96292753827953021781576062104, 19.30213092278224342229830572021, 20.00245974424076084351102627239, 20.7557159815822368058451821460, 21.65021805765247334772355119192, 22.46654453203862522017597231098, 23.31486809551704483008297881060

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