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Properties

Label 1-17e2-289.61-r1-0-0
Degree $1$
Conductor $289$
Sign $-0.232 - 0.972i$
Analytic cond. $31.0573$
Root an. cond. $31.0573$
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  + (0.873 − 0.486i)2-s + (0.955 − 0.295i)3-s + (0.526 − 0.850i)4-s + (0.940 + 0.339i)5-s + (0.690 − 0.723i)6-s + (−0.837 − 0.545i)7-s + (0.0461 − 0.998i)8-s + (0.824 − 0.565i)9-s + (0.986 − 0.160i)10-s + (0.811 − 0.584i)11-s + (0.251 − 0.967i)12-s + (−0.673 − 0.739i)13-s + (−0.997 − 0.0692i)14-s + (0.998 + 0.0461i)15-s + (−0.445 − 0.895i)16-s + ⋯
L(s)  = 1  + (0.873 − 0.486i)2-s + (0.955 − 0.295i)3-s + (0.526 − 0.850i)4-s + (0.940 + 0.339i)5-s + (0.690 − 0.723i)6-s + (−0.837 − 0.545i)7-s + (0.0461 − 0.998i)8-s + (0.824 − 0.565i)9-s + (0.986 − 0.160i)10-s + (0.811 − 0.584i)11-s + (0.251 − 0.967i)12-s + (−0.673 − 0.739i)13-s + (−0.997 − 0.0692i)14-s + (0.998 + 0.0461i)15-s + (−0.445 − 0.895i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.232 - 0.972i$
Analytic conductor: \(31.0573\)
Root analytic conductor: \(31.0573\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 289,\ (1:\ ),\ -0.232 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.197409759 - 4.053628591i\)
\(L(\frac12)\) \(\approx\) \(3.197409759 - 4.053628591i\)
\(L(1)\) \(\approx\) \(2.242107203 - 1.383549398i\)
\(L(1)\) \(\approx\) \(2.242107203 - 1.383549398i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + (0.873 - 0.486i)T \)
3 \( 1 + (0.955 - 0.295i)T \)
5 \( 1 + (0.940 + 0.339i)T \)
7 \( 1 + (-0.837 - 0.545i)T \)
11 \( 1 + (0.811 - 0.584i)T \)
13 \( 1 + (-0.673 - 0.739i)T \)
19 \( 1 + (-0.486 + 0.873i)T \)
23 \( 1 + (-0.545 + 0.837i)T \)
29 \( 1 + (0.986 + 0.160i)T \)
31 \( 1 + (-0.905 - 0.424i)T \)
37 \( 1 + (0.506 - 0.862i)T \)
41 \( 1 + (0.466 - 0.884i)T \)
43 \( 1 + (0.317 + 0.948i)T \)
47 \( 1 + (0.183 + 0.982i)T \)
53 \( 1 + (0.565 + 0.824i)T \)
59 \( 1 + (0.403 - 0.914i)T \)
61 \( 1 + (-0.723 - 0.690i)T \)
67 \( 1 + (0.273 + 0.961i)T \)
71 \( 1 + (-0.206 - 0.978i)T \)
73 \( 1 + (0.754 + 0.656i)T \)
79 \( 1 + (0.993 + 0.115i)T \)
83 \( 1 + (-0.638 + 0.769i)T \)
89 \( 1 + (-0.673 + 0.739i)T \)
97 \( 1 + (0.978 - 0.206i)T \)
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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

−25.53565467065208984576615412700, −24.757298307755782771009724769237, −24.03025384513251189252309656198, −22.5740973268781792173201341667, −21.8050128788566590805688347022, −21.38930197100268302422871905929, −20.17603693854872147039845995939, −19.58101004053316721949149916966, −18.16887156034293536056301985664, −16.92960593273631291258503513277, −16.26500590034695777447450958173, −15.158369766150664134138274106515, −14.46297225696949191133695913520, −13.61567463766705075972168641503, −12.789952108448194138377244622740, −11.98787442281421881454856196515, −10.24026202587250243491223330241, −9.25801474952613438347672327367, −8.59199968134015245417622857283, −7.03538320148389480338753008455, −6.3530775063877990110223153940, −4.97153825682394190336084981538, −4.11707121061320637980297221038, −2.73970476679973539738518815310, −2.01631767714024694655702243291, 1.03797523944799733209656181592, 2.259541691682608589355065309095, 3.22090415187378594505975815583, 4.057726342053375963616868548812, 5.75231320250406189438077840263, 6.530207882262379934612358794643, 7.58156768691002142337196902462, 9.27729119676624170180296831144, 9.90507119032734896075227090451, 10.83127728009559719428349159873, 12.407015202839397195086315843675, 12.99011826710038527508393652293, 14.04652752151386112417708115258, 14.31842680479775777292154049558, 15.47566662918883912872968326629, 16.64139188021886384525309990470, 17.92976956794992207380623813581, 19.11674507876085470948299383093, 19.6467196851080802658120734715, 20.49255076314507358555800408567, 21.48376628621103419230306910273, 22.14950524158740255047996927190, 23.0860564091052227379886413541, 24.1898898794800133658749948383, 25.095036214001171967926297602458

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