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Properties

Label 1-1037-1037.627-r1-0-0
Degree $1$
Conductor $1037$
Sign $-0.809 + 0.586i$
Analytic cond. $111.441$
Root an. cond. $111.441$
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  + (0.978 + 0.207i)2-s + (0.891 + 0.453i)3-s + (0.913 + 0.406i)4-s + (0.777 + 0.629i)5-s + (0.777 + 0.629i)6-s + (−0.998 − 0.0523i)7-s + (0.809 + 0.587i)8-s + (0.587 + 0.809i)9-s + (0.629 + 0.777i)10-s + (−0.707 + 0.707i)11-s + (0.629 + 0.777i)12-s + (0.5 − 0.866i)13-s + (−0.965 − 0.258i)14-s + (0.406 + 0.913i)15-s + (0.669 + 0.743i)16-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.891 + 0.453i)3-s + (0.913 + 0.406i)4-s + (0.777 + 0.629i)5-s + (0.777 + 0.629i)6-s + (−0.998 − 0.0523i)7-s + (0.809 + 0.587i)8-s + (0.587 + 0.809i)9-s + (0.629 + 0.777i)10-s + (−0.707 + 0.707i)11-s + (0.629 + 0.777i)12-s + (0.5 − 0.866i)13-s + (−0.965 − 0.258i)14-s + (0.406 + 0.913i)15-s + (0.669 + 0.743i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $-0.809 + 0.586i$
Analytic conductor: \(111.441\)
Root analytic conductor: \(111.441\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (1:\ ),\ -0.809 + 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.702222332 + 5.249139066i\)
\(L(\frac12)\) \(\approx\) \(1.702222332 + 5.249139066i\)
\(L(1)\) \(\approx\) \(2.143455473 + 1.426417115i\)
\(L(1)\) \(\approx\) \(2.143455473 + 1.426417115i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.978 + 0.207i)T \)
3 \( 1 + (0.891 + 0.453i)T \)
5 \( 1 + (0.777 + 0.629i)T \)
7 \( 1 + (-0.998 - 0.0523i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 + (-0.156 + 0.987i)T \)
29 \( 1 + (-0.258 + 0.965i)T \)
31 \( 1 + (-0.838 - 0.544i)T \)
37 \( 1 + (0.891 - 0.453i)T \)
41 \( 1 + (-0.891 + 0.453i)T \)
43 \( 1 + (-0.913 + 0.406i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (-0.994 + 0.104i)T \)
71 \( 1 + (-0.777 + 0.629i)T \)
73 \( 1 + (0.629 + 0.777i)T \)
79 \( 1 + (0.358 - 0.933i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (0.544 - 0.838i)T \)
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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

−21.0788655962032649189035552144, −20.46228712771915883480480059220, −19.66801052078104863759199383784, −18.84387117743630735015461729764, −18.35851476590853148701809674523, −16.69038829912141449164356124826, −16.37895156684288401588269060536, −15.37039281921112827104236459391, −14.52923363306349804892302105239, −13.531683463528109163018260883378, −13.430041157609130525216021623689, −12.60082371166783342206558240738, −11.903689856703534535820028053133, −10.54223244631673802118334314886, −9.88509374347334451156095896624, −8.9083911931306406866035912715, −8.162547761726694928692382644567, −6.83564015929463534679114732395, −6.29105813331834167471013412667, −5.477220485680980284257143253083, −4.224057666016974131855439903011, −3.48136715092972144881456284813, −2.45665239507958941832142309318, −1.8337662096876275977524656521, −0.588504498470886758576202404633, 1.74413060374610865018976539403, 2.68821530230732059188408320871, 3.19063747162287491386109281574, 4.110440175690396153848105189866, 5.23826758589319692147446409014, 5.97780772196950738800681726157, 7.02731636023045638739356594788, 7.59820815901995831859562115372, 8.80327912726201898398914613627, 9.817211254212799497485800322023, 10.42190379170298274944747069247, 11.17088317864644968715581537723, 12.72663256174548827097123222757, 13.16044521155328826546778809501, 13.63561935740734540192998724799, 14.775022322423332971207141122361, 15.14214572966760958358320007631, 15.88842474635932029336859157177, 16.690155764194600911721763674493, 17.76127239134879790420239884124, 18.65380695662137928283280332258, 19.71726223531764137071590499339, 20.23095251667146704523254358031, 21.03294559332706506307680058094, 21.82298280505430685476877125921

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