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Properties

Label 2-17-17.8-c3-0-1
Degree $2$
Conductor $17$
Sign $0.911 + 0.410i$
Analytic cond. $1.00303$
Root an. cond. $1.00151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.161 + 0.161i)2-s + (3.15 − 7.61i)3-s + 7.94i·4-s + (2.54 + 1.05i)5-s + (0.718 + 1.73i)6-s + (−19.8 + 8.23i)7-s + (−2.56 − 2.56i)8-s + (−28.9 − 28.9i)9-s + (−0.579 + 0.239i)10-s + (20.8 + 50.3i)11-s + (60.5 + 25.0i)12-s − 52.4i·13-s + (1.87 − 4.53i)14-s + (16.0 − 16.0i)15-s − 62.7·16-s + (33.5 − 61.5i)17-s + ⋯
L(s)  = 1  + (−0.0569 + 0.0569i)2-s + (0.606 − 1.46i)3-s + 0.993i·4-s + (0.227 + 0.0941i)5-s + (0.0488 + 0.118i)6-s + (−1.07 + 0.444i)7-s + (−0.113 − 0.113i)8-s + (−1.07 − 1.07i)9-s + (−0.0183 + 0.00758i)10-s + (0.572 + 1.38i)11-s + (1.45 + 0.602i)12-s − 1.11i·13-s + (0.0358 − 0.0864i)14-s + (0.275 − 0.275i)15-s − 0.980·16-s + (0.478 − 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.911 + 0.410i$
Analytic conductor: \(1.00303\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :3/2),\ 0.911 + 0.410i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.08914 - 0.233831i\)
\(L(\frac12)\) \(\approx\) \(1.08914 - 0.233831i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-33.5 + 61.5i)T \)
good2 \( 1 + (0.161 - 0.161i)T - 8iT^{2} \)
3 \( 1 + (-3.15 + 7.61i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (-2.54 - 1.05i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (19.8 - 8.23i)T + (242. - 242. i)T^{2} \)
11 \( 1 + (-20.8 - 50.3i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + 52.4iT - 2.19e3T^{2} \)
19 \( 1 + (-13.8 + 13.8i)T - 6.85e3iT^{2} \)
23 \( 1 + (5.33 + 12.8i)T + (-8.60e3 + 8.60e3i)T^{2} \)
29 \( 1 + (-64.6 - 26.7i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + (63.9 - 154. i)T + (-2.10e4 - 2.10e4i)T^{2} \)
37 \( 1 + (76.0 - 183. i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-401. + 166. i)T + (4.87e4 - 4.87e4i)T^{2} \)
43 \( 1 + (-89.9 - 89.9i)T + 7.95e4iT^{2} \)
47 \( 1 - 207. iT - 1.03e5T^{2} \)
53 \( 1 + (220. - 220. i)T - 1.48e5iT^{2} \)
59 \( 1 + (407. + 407. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-72.4 + 29.9i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 - 359.T + 3.00e5T^{2} \)
71 \( 1 + (-81.5 + 196. i)T + (-2.53e5 - 2.53e5i)T^{2} \)
73 \( 1 + (-26.8 - 11.1i)T + (2.75e5 + 2.75e5i)T^{2} \)
79 \( 1 + (-327. - 790. i)T + (-3.48e5 + 3.48e5i)T^{2} \)
83 \( 1 + (-9.67 + 9.67i)T - 5.71e5iT^{2} \)
89 \( 1 - 651. iT - 7.04e5T^{2} \)
97 \( 1 + (1.10e3 + 458. i)T + (6.45e5 + 6.45e5i)T^{2} \)
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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

−18.21839307991142742185535940248, −17.51066461239848595441339818998, −15.77742404286221559423046356837, −14.01395537089286119703596445981, −12.66865075572563584507152977112, −12.30467969059534034536152254567, −9.411861261245147617561087722200, −7.80911293890994938460299409290, −6.69215851517051371857735440021, −2.80811572346532257613209573940, 3.83434324346232986494305054609, 5.99428858372868210881898298167, 9.061976897313907715867345366026, 9.857099337988617231862647489296, 11.07807009325617276065682987141, 13.69616040899096529874629439079, 14.60012879705141850551372015477, 15.94446122965336547875870431583, 16.71533390350133679044071498810, 19.11953457980084304618984746641

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