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Properties

Label 1-1157-1157.310-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

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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: $\chi_{1157} (310, \cdot )$
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 \( 1 + (-0.0950 - 0.995i)T \)
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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

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

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