Properties

Label 1-1157-1157.474-r0-0-0
Degree $1$
Conductor $1157$
Sign $-0.605 - 0.795i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.618 + 0.786i)2-s + (−0.520 + 0.853i)3-s + (−0.235 + 0.971i)4-s + (−0.415 − 0.909i)5-s + (−0.992 + 0.118i)6-s + (0.771 − 0.636i)7-s + (−0.909 + 0.415i)8-s + (−0.458 − 0.888i)9-s + (0.458 − 0.888i)10-s + (0.0950 + 0.995i)11-s + (−0.707 − 0.707i)12-s + (0.977 + 0.212i)14-s + (0.992 + 0.118i)15-s + (−0.888 − 0.458i)16-s + (0.618 − 0.786i)17-s + (0.415 − 0.909i)18-s + ⋯
L(s)  = 1  + (0.618 + 0.786i)2-s + (−0.520 + 0.853i)3-s + (−0.235 + 0.971i)4-s + (−0.415 − 0.909i)5-s + (−0.992 + 0.118i)6-s + (0.771 − 0.636i)7-s + (−0.909 + 0.415i)8-s + (−0.458 − 0.888i)9-s + (0.458 − 0.888i)10-s + (0.0950 + 0.995i)11-s + (−0.707 − 0.707i)12-s + (0.977 + 0.212i)14-s + (0.992 + 0.118i)15-s + (−0.888 − 0.458i)16-s + (0.618 − 0.786i)17-s + (0.415 − 0.909i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (474, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ -0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1998447002 + 0.4034252527i\)
\(L(\frac12)\) \(\approx\) \(-0.1998447002 + 0.4034252527i\)
\(L(1)\) \(\approx\) \(0.7143357083 + 0.5457298236i\)
\(L(1)\) \(\approx\) \(0.7143357083 + 0.5457298236i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.618 + 0.786i)T \)
3 \( 1 + (-0.520 + 0.853i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (0.771 - 0.636i)T \)
11 \( 1 + (0.0950 + 0.995i)T \)
17 \( 1 + (0.618 - 0.786i)T \)
19 \( 1 + (-0.952 - 0.304i)T \)
23 \( 1 + (-0.739 + 0.672i)T \)
29 \( 1 + (0.165 + 0.986i)T \)
31 \( 1 + (-0.977 - 0.212i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (-0.999 + 0.0237i)T \)
43 \( 1 + (-0.165 + 0.986i)T \)
47 \( 1 + (-0.959 + 0.281i)T \)
53 \( 1 + (-0.281 + 0.959i)T \)
59 \( 1 + (-0.853 + 0.520i)T \)
61 \( 1 + (-0.828 - 0.560i)T \)
67 \( 1 + (0.971 - 0.235i)T \)
71 \( 1 + (-0.580 - 0.814i)T \)
73 \( 1 + (-0.540 - 0.841i)T \)
79 \( 1 + (0.540 + 0.841i)T \)
83 \( 1 + (-0.599 - 0.800i)T \)
97 \( 1 + (-0.0950 + 0.995i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.97625552201258975976124294126, −19.80874730785455277657529762549, −19.09619360317226148860267874552, −18.68723279337206518478403004895, −18.077733412906150412625632383460, −17.13874753347587150664882684879, −16.027050495124324185650688669275, −14.9790959020001543142539014415, −14.40157890441786466571507350483, −13.77312833489963754829330516667, −12.71679904647893200002264213089, −12.09072841846364954937563621402, −11.4022937691955449206037139627, −10.845556234597045728949690001599, −10.15431577616300961036935023883, −8.590126877573993196931052620617, −8.06367601710325018184830279791, −6.81084808445984341699586472771, −6.00819853594357721423077668737, −5.503225340458374699563557652394, −4.26314293537035998633265750466, −3.331059431990759067313225188094, −2.28273341524059452492626168008, −1.65836291215662961759616916462, −0.152727053609296844839855283325, 1.53568026597019196125979714174, 3.27257368283518699752345858545, 4.14383181334581950589648021411, 4.80952214037434501654803774900, 5.165762294935666556851269901696, 6.34089169252630716013310346225, 7.34662480688759034218574564786, 8.05807320980136980584503223386, 9.002597244102272520909178666051, 9.75685986702115330251927334315, 10.90100454104416834747121101207, 11.79127226272493023086900157378, 12.29720032245959963002213561023, 13.23655867910032393392950727557, 14.217351562247636537405939868062, 14.95351180217256755818452739037, 15.54846010944395200803562133942, 16.4363878182557011522245531165, 16.87879725562356013545832567302, 17.560471941309428560136355726393, 18.260863409572509346413997436561, 19.99929290851287238877464873265, 20.38339186214227392286041245734, 21.14610393677389187232325783696, 21.75841506776597211273774885605

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