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Properties

Label 1-517-517.115-r0-0-0
Degree $1$
Conductor $517$
Sign $-0.867 + 0.497i$
Analytic cond. $2.40093$
Root an. cond. $2.40093$
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  + (−0.0136 − 0.999i)2-s + (0.360 − 0.932i)3-s + (−0.999 + 0.0273i)4-s + (−0.122 − 0.992i)5-s + (−0.937 − 0.347i)6-s + (0.986 − 0.163i)7-s + (0.0409 + 0.999i)8-s + (−0.740 − 0.672i)9-s + (−0.990 + 0.136i)10-s + (−0.334 + 0.942i)12-s + (0.507 − 0.861i)13-s + (−0.176 − 0.984i)14-s + (−0.969 − 0.243i)15-s + (0.998 − 0.0546i)16-s + (−0.385 − 0.922i)17-s + (−0.662 + 0.749i)18-s + ⋯
L(s)  = 1  + (−0.0136 − 0.999i)2-s + (0.360 − 0.932i)3-s + (−0.999 + 0.0273i)4-s + (−0.122 − 0.992i)5-s + (−0.937 − 0.347i)6-s + (0.986 − 0.163i)7-s + (0.0409 + 0.999i)8-s + (−0.740 − 0.672i)9-s + (−0.990 + 0.136i)10-s + (−0.334 + 0.942i)12-s + (0.507 − 0.861i)13-s + (−0.176 − 0.984i)14-s + (−0.969 − 0.243i)15-s + (0.998 − 0.0546i)16-s + (−0.385 − 0.922i)17-s + (−0.662 + 0.749i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.867 + 0.497i$
Analytic conductor: \(2.40093\)
Root analytic conductor: \(2.40093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (0:\ ),\ -0.867 + 0.497i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3470943802 - 1.303584751i\)
\(L(\frac12)\) \(\approx\) \(-0.3470943802 - 1.303584751i\)
\(L(1)\) \(\approx\) \(0.5044947624 - 0.9932333370i\)
\(L(1)\) \(\approx\) \(0.5044947624 - 0.9932333370i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.0136 - 0.999i)T \)
3 \( 1 + (0.360 - 0.932i)T \)
5 \( 1 + (-0.122 - 0.992i)T \)
7 \( 1 + (0.986 - 0.163i)T \)
13 \( 1 + (0.507 - 0.861i)T \)
17 \( 1 + (-0.385 - 0.922i)T \)
19 \( 1 + (0.905 + 0.423i)T \)
23 \( 1 + (-0.576 - 0.816i)T \)
29 \( 1 + (-0.662 + 0.749i)T \)
31 \( 1 + (0.256 - 0.966i)T \)
37 \( 1 + (-0.176 + 0.984i)T \)
41 \( 1 + (0.554 + 0.832i)T \)
43 \( 1 + (-0.334 - 0.942i)T \)
53 \( 1 + (-0.620 + 0.784i)T \)
59 \( 1 + (0.792 + 0.609i)T \)
61 \( 1 + (0.721 + 0.692i)T \)
67 \( 1 + (0.460 - 0.887i)T \)
71 \( 1 + (-0.0136 + 0.999i)T \)
73 \( 1 + (-0.868 - 0.496i)T \)
79 \( 1 + (0.641 + 0.767i)T \)
83 \( 1 + (-0.996 - 0.0818i)T \)
89 \( 1 + (0.682 + 0.730i)T \)
97 \( 1 + (0.998 + 0.0546i)T \)
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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

−24.040635689119515216169292471561, −23.26741582683554448026693657226, −22.30968759238875670392060971596, −21.68490599803730338294348714846, −21.03149680531422609404485274713, −19.671605523562198286273409668973, −18.894814548472424656940695057548, −17.83338882979929933247989781725, −17.3384480276163321313685390628, −16.000752361803949052313723781913, −15.61648333989115436802488045057, −14.57210384274113398110860927750, −14.27716320995451474413658979347, −13.37444758063058883250832169095, −11.66162612454805415917735618250, −10.934886226691222220620768648414, −9.92011732563659040365811406074, −9.02473830120805018673106538383, −8.1510924462942388999562581104, −7.350694197306631904654894033617, −6.18299774970713827006369366380, −5.26433274762714495790370875982, −4.198302110964395399197006976574, −3.49713585663181183867258931996, −1.96107481599120084517994103668, 0.75171123615359448168331040559, 1.52969124527105020301306066745, 2.62205128884463010224149202686, 3.823029563843431103829160386154, 4.92623967289236147470176101948, 5.775849195289205567875038815455, 7.508850282826140336282856035848, 8.224874900661114307851305445549, 8.86109416850982936921255126509, 9.921027489239988612616798339574, 11.226368016071833023346699111967, 11.8416631136837510229072978546, 12.64915280174077283898885214891, 13.45958389905822050924111203172, 14.05041285171640279794506614610, 15.11062498784301620157416896360, 16.52315137767024747713600628791, 17.52481197861072083279367066772, 18.15507352471667530396375703752, 18.79494657779938120059342065464, 20.163418789244155137182769311908, 20.31004257712552878590231383869, 20.88560577418876737822353184860, 22.23651414843385592142469511905, 23.070456536056265381561152894013

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