Properties

Label 2-17-17.16-c3-0-3
Degree $2$
Conductor $17$
Sign $0.973 + 0.230i$
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  + 2.37·2-s − 4.44i·3-s − 2.37·4-s + 19.4i·5-s − 10.5i·6-s − 14.9i·7-s − 24.6·8-s + 7.23·9-s + 46.1i·10-s − 31.1i·11-s + 10.5i·12-s − 5.21·13-s − 35.5i·14-s + 86.4·15-s − 39.3·16-s + (68.2 + 16.1i)17-s + ⋯
L(s)  = 1  + 0.838·2-s − 0.855i·3-s − 0.296·4-s + 1.73i·5-s − 0.717i·6-s − 0.809i·7-s − 1.08·8-s + 0.267·9-s + 1.45i·10-s − 0.853i·11-s + 0.253i·12-s − 0.111·13-s − 0.678i·14-s + 1.48·15-s − 0.615·16-s + (0.973 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(1.00303\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :3/2),\ 0.973 + 0.230i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.31095 - 0.152878i\)
\(L(\frac12)\) \(\approx\) \(1.31095 - 0.152878i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-68.2 - 16.1i)T \)
good2 \( 1 - 2.37T + 8T^{2} \)
3 \( 1 + 4.44iT - 27T^{2} \)
5 \( 1 - 19.4iT - 125T^{2} \)
7 \( 1 + 14.9iT - 343T^{2} \)
11 \( 1 + 31.1iT - 1.33e3T^{2} \)
13 \( 1 + 5.21T + 2.19e3T^{2} \)
19 \( 1 + 28T + 6.85e3T^{2} \)
23 \( 1 - 167. iT - 1.21e4T^{2} \)
29 \( 1 + 136. iT - 2.43e4T^{2} \)
31 \( 1 - 50.5iT - 2.97e4T^{2} \)
37 \( 1 - 260. iT - 5.06e4T^{2} \)
41 \( 1 + 183. iT - 6.89e4T^{2} \)
43 \( 1 + 348.T + 7.95e4T^{2} \)
47 \( 1 - 318.T + 1.03e5T^{2} \)
53 \( 1 + 408.T + 1.48e5T^{2} \)
59 \( 1 - 108.T + 2.05e5T^{2} \)
61 \( 1 - 123. iT - 2.26e5T^{2} \)
67 \( 1 + 243.T + 3.00e5T^{2} \)
71 \( 1 + 42.7iT - 3.57e5T^{2} \)
73 \( 1 + 875. iT - 3.89e5T^{2} \)
79 \( 1 + 750. iT - 4.93e5T^{2} \)
83 \( 1 + 472.T + 5.71e5T^{2} \)
89 \( 1 - 376.T + 7.04e5T^{2} \)
97 \( 1 - 303. iT - 9.12e5T^{2} \)
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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

−18.59641338019204734025794913016, −17.43735411737250900906516537789, −15.25848146939513783510658829590, −14.03930960131214703925058849590, −13.42761283186875752180698191079, −11.75212381344530589661965658315, −10.19791111715273864048033247342, −7.56594401686642402426066623063, −6.22065577514056024710167293684, −3.46918795984664353324196041812, 4.37923220276666023327149617093, 5.31977091221502005946431867360, 8.699953750691681295823385858616, 9.709340471103488431391009543474, 12.28563232661045129803580875266, 12.80989534907016739716398128188, 14.61747177278034738770858406032, 15.75378030900008580055718880191, 16.84098036077514651031132246251, 18.42330567947216344792576846589

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