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Properties

Label 1-517-517.97-r0-0-0
Degree $1$
Conductor $517$
Sign $-0.528 + 0.849i$
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  + (−0.894 + 0.447i)2-s + (−0.999 + 0.0273i)3-s + (0.598 − 0.800i)4-s + (0.507 − 0.861i)5-s + (0.881 − 0.472i)6-s + (0.758 + 0.652i)7-s + (−0.176 + 0.984i)8-s + (0.998 − 0.0546i)9-s + (−0.0682 + 0.997i)10-s + (−0.576 + 0.816i)12-s + (−0.740 + 0.672i)13-s + (−0.969 − 0.243i)14-s + (−0.484 + 0.874i)15-s + (−0.282 − 0.959i)16-s + (−0.620 + 0.784i)17-s + (−0.868 + 0.496i)18-s + ⋯
L(s)  = 1  + (−0.894 + 0.447i)2-s + (−0.999 + 0.0273i)3-s + (0.598 − 0.800i)4-s + (0.507 − 0.861i)5-s + (0.881 − 0.472i)6-s + (0.758 + 0.652i)7-s + (−0.176 + 0.984i)8-s + (0.998 − 0.0546i)9-s + (−0.0682 + 0.997i)10-s + (−0.576 + 0.816i)12-s + (−0.740 + 0.672i)13-s + (−0.969 − 0.243i)14-s + (−0.484 + 0.874i)15-s + (−0.282 − 0.959i)16-s + (−0.620 + 0.784i)17-s + (−0.868 + 0.496i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.528 + 0.849i$
Analytic conductor: \(2.40093\)
Root analytic conductor: \(2.40093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (0:\ ),\ -0.528 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2245275737 + 0.4040696707i\)
\(L(\frac12)\) \(\approx\) \(0.2245275737 + 0.4040696707i\)
\(L(1)\) \(\approx\) \(0.5165436223 + 0.1593687581i\)
\(L(1)\) \(\approx\) \(0.5165436223 + 0.1593687581i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.894 + 0.447i)T \)
3 \( 1 + (-0.999 + 0.0273i)T \)
5 \( 1 + (0.507 - 0.861i)T \)
7 \( 1 + (0.758 + 0.652i)T \)
13 \( 1 + (-0.740 + 0.672i)T \)
17 \( 1 + (-0.620 + 0.784i)T \)
19 \( 1 + (-0.662 + 0.749i)T \)
23 \( 1 + (0.460 + 0.887i)T \)
29 \( 1 + (-0.868 + 0.496i)T \)
31 \( 1 + (0.824 - 0.565i)T \)
37 \( 1 + (-0.969 + 0.243i)T \)
41 \( 1 + (-0.435 + 0.900i)T \)
43 \( 1 + (-0.576 - 0.816i)T \)
53 \( 1 + (0.721 + 0.692i)T \)
59 \( 1 + (-0.955 - 0.295i)T \)
61 \( 1 + (0.641 + 0.767i)T \)
67 \( 1 + (0.854 + 0.519i)T \)
71 \( 1 + (-0.894 - 0.447i)T \)
73 \( 1 + (0.360 - 0.932i)T \)
79 \( 1 + (-0.122 - 0.992i)T \)
83 \( 1 + (-0.937 + 0.347i)T \)
89 \( 1 + (-0.917 + 0.398i)T \)
97 \( 1 + (-0.282 + 0.959i)T \)
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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

−22.993114319504790257098863756144, −22.38951461343604267526770182749, −21.51643630166837706779468343691, −20.83040122467124521007267058585, −19.80276950261971790683154926294, −18.783939329505132524910562193907, −18.08504920321246549399817363729, −17.31903845923785207070056194475, −17.08264879617503359527625980337, −15.76773956289728303421485387501, −14.91924844767044621333840393886, −13.62548688753803314776571130553, −12.70253049697510775689978176687, −11.611728939719698877170686283811, −10.94322743013157459737726912689, −10.3918636105112181147701357691, −9.58501847150447424212789819503, −8.254813846949733895338492190106, −7.03296809888920913003459268682, −6.82573031061459507553426662998, −5.338459489024721945036587127117, −4.2502485283339592585395823667, −2.78865804522816774901766581569, −1.78169242845894977589907478297, −0.37919690168367970043955338713, 1.432564522530839344707662782906, 2.02129484680607502610937202961, 4.417252775745732030284623227512, 5.27760177337153796251110854994, 5.949224061782200751643012439602, 6.93219898331369920598131520527, 8.08479500028797175496705848304, 8.9458093754953356614894109416, 9.78138918286992235255379824094, 10.69789594103460624129906350967, 11.66553572401544247325536127427, 12.305931658137633044523720601, 13.50027581356919874514724112224, 14.81815212067174386697694037157, 15.47646756460065456058135051, 16.58178256210474242085232372199, 17.11113186010879097263567974613, 17.63380190258266151853096269308, 18.57590375338845392739353210854, 19.32471572281648302888877345680, 20.54664672557563683609429652864, 21.3284449629547150274930133891, 21.99439438418032672094806592217, 23.37654397459223830893253623945, 24.0707996985980131501897004515

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