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Properties

Label 2-17-17.13-c3-0-3
Degree $2$
Conductor $17$
Sign $-0.0198 + 0.999i$
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  − 5.46i·2-s + (3.94 + 3.94i)3-s − 21.8·4-s + (4.79 + 4.79i)5-s + (21.5 − 21.5i)6-s + (3.33 − 3.33i)7-s + 75.9i·8-s + 4.13i·9-s + (26.1 − 26.1i)10-s + (−6.70 + 6.70i)11-s + (−86.3 − 86.3i)12-s − 33.7·13-s + (−18.2 − 18.2i)14-s + 37.8i·15-s + 240.·16-s + (−56.0 + 42.0i)17-s + ⋯
L(s)  = 1  − 1.93i·2-s + (0.759 + 0.759i)3-s − 2.73·4-s + (0.428 + 0.428i)5-s + (1.46 − 1.46i)6-s + (0.180 − 0.180i)7-s + 3.35i·8-s + 0.153i·9-s + (0.828 − 0.828i)10-s + (−0.183 + 0.183i)11-s + (−2.07 − 2.07i)12-s − 0.719·13-s + (−0.348 − 0.348i)14-s + 0.650i·15-s + 3.75·16-s + (−0.800 + 0.599i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0198 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0198 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.0198 + 0.999i$
Analytic conductor: \(1.00303\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :3/2),\ -0.0198 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.780798 - 0.796458i\)
\(L(\frac12)\) \(\approx\) \(0.780798 - 0.796458i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (56.0 - 42.0i)T \)
good2 \( 1 + 5.46iT - 8T^{2} \)
3 \( 1 + (-3.94 - 3.94i)T + 27iT^{2} \)
5 \( 1 + (-4.79 - 4.79i)T + 125iT^{2} \)
7 \( 1 + (-3.33 + 3.33i)T - 343iT^{2} \)
11 \( 1 + (6.70 - 6.70i)T - 1.33e3iT^{2} \)
13 \( 1 + 33.7T + 2.19e3T^{2} \)
19 \( 1 + 27.4iT - 6.85e3T^{2} \)
23 \( 1 + (-58.6 + 58.6i)T - 1.21e4iT^{2} \)
29 \( 1 + (-147. - 147. i)T + 2.43e4iT^{2} \)
31 \( 1 + (158. + 158. i)T + 2.97e4iT^{2} \)
37 \( 1 + (-122. - 122. i)T + 5.06e4iT^{2} \)
41 \( 1 + (60.9 - 60.9i)T - 6.89e4iT^{2} \)
43 \( 1 + 258. iT - 7.95e4T^{2} \)
47 \( 1 - 88.9T + 1.03e5T^{2} \)
53 \( 1 - 541. iT - 1.48e5T^{2} \)
59 \( 1 + 13.5iT - 2.05e5T^{2} \)
61 \( 1 + (112. - 112. i)T - 2.26e5iT^{2} \)
67 \( 1 - 357.T + 3.00e5T^{2} \)
71 \( 1 + (-679. - 679. i)T + 3.57e5iT^{2} \)
73 \( 1 + (635. + 635. i)T + 3.89e5iT^{2} \)
79 \( 1 + (319. - 319. i)T - 4.93e5iT^{2} \)
83 \( 1 + 559. iT - 5.71e5T^{2} \)
89 \( 1 + 602.T + 7.04e5T^{2} \)
97 \( 1 + (580. + 580. i)T + 9.12e5iT^{2} \)
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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

−18.53003553820768468077859825068, −17.40819948824144715890009167572, −14.90070956886864669638298070087, −13.88654922436473567943511993813, −12.50734181003365387266770727976, −10.85959743151674030133685033941, −9.907028765555727283907860338509, −8.760918225907517778717444941318, −4.41212984817514497884039459901, −2.67027849164515275936211708830, 5.17691914227545174080496313913, 6.99890415722263368711418769673, 8.199066667541582882068144768172, 9.354151487172606028043634384289, 12.91319481299162945445977192766, 13.76829597601024028390081096306, 14.79687087496240026472270154159, 16.12907309338970932001497368690, 17.34682480188161734102095648336, 18.33153098567529717425155377158

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