Properties

Label 2-17-17.5-c2-0-1
Degree $2$
Conductor $17$
Sign $0.581 + 0.813i$
Analytic cond. $0.463216$
Root an. cond. $0.680600$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.841 − 2.03i)2-s + (0.0897 + 0.134i)3-s + (−0.590 + 0.590i)4-s + (−1.04 + 5.26i)5-s + (0.197 − 0.295i)6-s + (1.21 + 6.12i)7-s + (−6.42 − 2.66i)8-s + (3.43 − 8.29i)9-s + (11.5 − 2.30i)10-s + (−12.1 − 8.11i)11-s + (−0.132 − 0.0263i)12-s + (4.79 + 4.79i)13-s + (11.4 − 7.62i)14-s + (−0.801 + 0.331i)15-s + 18.6i·16-s + (6.50 + 15.7i)17-s + ⋯
L(s)  = 1  + (−0.420 − 1.01i)2-s + (0.0299 + 0.0447i)3-s + (−0.147 + 0.147i)4-s + (−0.209 + 1.05i)5-s + (0.0329 − 0.0492i)6-s + (0.174 + 0.874i)7-s + (−0.803 − 0.332i)8-s + (0.381 − 0.921i)9-s + (1.15 − 0.230i)10-s + (−1.10 − 0.737i)11-s + (−0.0110 − 0.00219i)12-s + (0.369 + 0.369i)13-s + (0.815 − 0.544i)14-s + (−0.0534 + 0.0221i)15-s + 1.16i·16-s + (0.382 + 0.923i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(0.463216\)
Root analytic conductor: \(0.680600\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :1),\ 0.581 + 0.813i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.618160 - 0.317877i\)
\(L(\frac12)\) \(\approx\) \(0.618160 - 0.317877i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-6.50 - 15.7i)T \)
good2 \( 1 + (0.841 + 2.03i)T + (-2.82 + 2.82i)T^{2} \)
3 \( 1 + (-0.0897 - 0.134i)T + (-3.44 + 8.31i)T^{2} \)
5 \( 1 + (1.04 - 5.26i)T + (-23.0 - 9.56i)T^{2} \)
7 \( 1 + (-1.21 - 6.12i)T + (-45.2 + 18.7i)T^{2} \)
11 \( 1 + (12.1 + 8.11i)T + (46.3 + 111. i)T^{2} \)
13 \( 1 + (-4.79 - 4.79i)T + 169iT^{2} \)
19 \( 1 + (9.56 + 23.0i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (-7.27 + 10.8i)T + (-202. - 488. i)T^{2} \)
29 \( 1 + (-32.3 - 6.44i)T + (776. + 321. i)T^{2} \)
31 \( 1 + (1.00 - 0.674i)T + (367. - 887. i)T^{2} \)
37 \( 1 + (25.8 + 38.6i)T + (-523. + 1.26e3i)T^{2} \)
41 \( 1 + (-6.13 - 30.8i)T + (-1.55e3 + 643. i)T^{2} \)
43 \( 1 + (27.8 - 67.3i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + (10.4 + 10.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (1.97 + 4.77i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-26.1 - 10.8i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (81.0 - 16.1i)T + (3.43e3 - 1.42e3i)T^{2} \)
67 \( 1 + 44.5iT - 4.48e3T^{2} \)
71 \( 1 + (-32.1 - 48.1i)T + (-1.92e3 + 4.65e3i)T^{2} \)
73 \( 1 + (0.262 - 1.32i)T + (-4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (24.7 + 16.5i)T + (2.38e3 + 5.76e3i)T^{2} \)
83 \( 1 + (62.2 - 25.7i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-90.1 + 90.1i)T - 7.92e3iT^{2} \)
97 \( 1 + (66.3 + 13.1i)T + (8.69e3 + 3.60e3i)T^{2} \)
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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

−18.60753229988716760539972469048, −18.03206053020274594876635796153, −15.66495948809099481420889097149, −14.80777743892533505890195500016, −12.72674105458849979929443727370, −11.36300872652885777138707335276, −10.40835665577570428983083355108, −8.779639353973859345825606974740, −6.39279628524038001392938270385, −2.95916334263223198660524035431, 5.06351843953026264270071707092, 7.39486115882744682103638689916, 8.318719143872788158483086939533, 10.28207272462103928356498155407, 12.34539181456178569352563527107, 13.72158070177151006942836165839, 15.53231800887825229265193031966, 16.37901499486708891759320594761, 17.27034132980581125914538385480, 18.61816235830431312333070580805

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