Properties

Label 2-17-17.15-c3-0-0
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: $\chi_{17} (15, \cdot )$
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

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

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