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Properties

Label 2-17-17.16-c15-0-14
Degree $2$
Conductor $17$
Sign $0.858 - 0.512i$
Analytic cond. $24.2578$
Root an. cond. $4.92523$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 326.·2-s − 2.24e3i·3-s + 7.36e4·4-s + 2.24e5i·5-s − 7.30e5i·6-s + 1.90e6i·7-s + 1.33e7·8-s + 9.32e6·9-s + 7.33e7i·10-s + 3.89e7i·11-s − 1.65e8i·12-s − 1.61e8·13-s + 6.19e8i·14-s + 5.03e8·15-s + 1.93e9·16-s + (1.45e9 − 8.66e8i)17-s + ⋯
L(s)  = 1  + 1.80·2-s − 0.591i·3-s + 2.24·4-s + 1.28i·5-s − 1.06i·6-s + 0.872i·7-s + 2.24·8-s + 0.650·9-s + 2.31i·10-s + 0.602i·11-s − 1.32i·12-s − 0.713·13-s + 1.57i·14-s + 0.761·15-s + 1.80·16-s + (0.858 − 0.512i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.858 - 0.512i$
Analytic conductor: \(24.2578\)
Root analytic conductor: \(4.92523\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :15/2),\ 0.858 - 0.512i)\)

Particular Values

\(L(8)\) \(\approx\) \(6.030857737\)
\(L(\frac12)\) \(\approx\) \(6.030857737\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.45e9 + 8.66e8i)T \)
good2 \( 1 - 326.T + 3.27e4T^{2} \)
3 \( 1 + 2.24e3iT - 1.43e7T^{2} \)
5 \( 1 - 2.24e5iT - 3.05e10T^{2} \)
7 \( 1 - 1.90e6iT - 4.74e12T^{2} \)
11 \( 1 - 3.89e7iT - 4.17e15T^{2} \)
13 \( 1 + 1.61e8T + 5.11e16T^{2} \)
19 \( 1 - 1.70e9T + 1.51e19T^{2} \)
23 \( 1 + 2.85e10iT - 2.66e20T^{2} \)
29 \( 1 + 2.44e10iT - 8.62e21T^{2} \)
31 \( 1 - 2.75e11iT - 2.34e22T^{2} \)
37 \( 1 - 5.05e10iT - 3.33e23T^{2} \)
41 \( 1 + 1.19e12iT - 1.55e24T^{2} \)
43 \( 1 + 9.60e11T + 3.17e24T^{2} \)
47 \( 1 + 1.33e12T + 1.20e25T^{2} \)
53 \( 1 + 1.42e13T + 7.31e25T^{2} \)
59 \( 1 - 3.43e13T + 3.65e26T^{2} \)
61 \( 1 + 3.02e13iT - 6.02e26T^{2} \)
67 \( 1 + 8.15e13T + 2.46e27T^{2} \)
71 \( 1 + 1.45e13iT - 5.87e27T^{2} \)
73 \( 1 - 7.56e12iT - 8.90e27T^{2} \)
79 \( 1 + 2.54e14iT - 2.91e28T^{2} \)
83 \( 1 - 1.50e14T + 6.11e28T^{2} \)
89 \( 1 + 3.18e14T + 1.74e29T^{2} \)
97 \( 1 - 9.45e14iT - 6.33e29T^{2} \)
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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

−14.84060902799546180003653737664, −14.15048214469853668095258639636, −12.61782530608958091773669371700, −11.94458308839524855739307058496, −10.34081688486697533497338124752, −7.30162850591679009965555427860, −6.46122436173923843237014929840, −4.90339260835715152284704461738, −3.10013032326691276885144255952, −2.10675694517766282358331810080, 1.30655157747277102605595633127, 3.55705611278455970896714645673, 4.54086785476175580412229710422, 5.58505355873626104733890548921, 7.51776648301407033685987023177, 9.837092501510907203783014984631, 11.49390152049023548652316132944, 12.78934385246879192296509552022, 13.58219417353255317573917155353, 14.96329486189237143862846159820

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