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Properties

Label 2-17-17.8-c1-0-0
Degree $2$
Conductor $17$
Sign $0.673 - 0.739i$
Analytic cond. $0.135745$
Root an. cond. $0.368436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 + 1.70i)2-s + (0.414 − i)3-s − 3.82i·4-s + (−0.707 − 0.292i)5-s + (1 + 2.41i)6-s + (−2.41 + i)7-s + (3.12 + 3.12i)8-s + (1.29 + 1.29i)9-s + (1.70 − 0.707i)10-s + (−1 − 2.41i)11-s + (−3.82 − 1.58i)12-s + 1.41i·13-s + (2.41 − 5.82i)14-s + (−0.585 + 0.585i)15-s − 2.99·16-s + (2.82 + 3i)17-s + ⋯
L(s)  = 1  + (−1.20 + 1.20i)2-s + (0.239 − 0.577i)3-s − 1.91i·4-s + (−0.316 − 0.130i)5-s + (0.408 + 0.985i)6-s + (−0.912 + 0.377i)7-s + (1.10 + 1.10i)8-s + (0.430 + 0.430i)9-s + (0.539 − 0.223i)10-s + (−0.301 − 0.727i)11-s + (−1.10 − 0.457i)12-s + 0.392i·13-s + (0.645 − 1.55i)14-s + (−0.151 + 0.151i)15-s − 0.749·16-s + (0.685 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.673 - 0.739i$
Analytic conductor: \(0.135745\)
Root analytic conductor: \(0.368436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :1/2),\ 0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324898 + 0.143520i\)
\(L(\frac12)\) \(\approx\) \(0.324898 + 0.143520i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-2.82 - 3i)T \)
good2 \( 1 + (1.70 - 1.70i)T - 2iT^{2} \)
3 \( 1 + (-0.414 + i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.707 + 0.292i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.41 - i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1 + 2.41i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.41iT - 13T^{2} \)
19 \( 1 + (-0.585 + 0.585i)T - 19iT^{2} \)
23 \( 1 + (1.82 + 4.41i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.292 + 0.121i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (3 - 7.24i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-3.53 + 8.53i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.12 + 0.464i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.585 + 0.585i)T + 43iT^{2} \)
47 \( 1 - 5.17iT - 47T^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + (-4.24 - 4.24i)T + 59iT^{2} \)
61 \( 1 + (3.53 - 1.46i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 + (2.07 - 5i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (11.9 + 4.94i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.82 - 4.41i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-8.24 + 8.24i)T - 83iT^{2} \)
89 \( 1 + 6.58iT - 89T^{2} \)
97 \( 1 + (-9.53 - 3.94i)T + (68.5 + 68.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.01262672139491597150907330135, −18.03589364335640505922420846295, −16.41162454944289048444564579122, −15.92403802772579260842764391622, −14.29692826949536558292876311602, −12.64612980251574111590139919250, −10.31660880773855664947822962414, −8.787455779585150857889124623441, −7.59353250379502107111273587820, −6.14743572620168908564155996007, 3.48616011786062289576592162458, 7.58544203495244752743489520942, 9.538072377630503931469175384201, 10.06336598180932952144978764529, 11.69015461759250798597364696764, 13.01438382222935165276952421923, 15.29053841161705035577093266686, 16.58309121534504542541749243770, 17.97365516705076145075471799876, 19.01463016934513832809466640220

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