Properties

Label 2-17e2-289.118-c1-0-6
Degree $2$
Conductor $289$
Sign $0.974 - 0.222i$
Analytic cond. $2.30767$
Root an. cond. $1.51910$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 − 1.19i)2-s + (−0.263 + 1.41i)3-s + (1.41 + 2.83i)4-s + (3.28 − 0.304i)5-s + (2.19 − 2.41i)6-s + (0.242 − 0.625i)7-s + (0.245 − 2.65i)8-s + (0.876 + 0.339i)9-s + (−6.71 − 3.34i)10-s + (−3.96 + 1.97i)11-s + (−4.37 + 1.24i)12-s + (−0.0517 + 0.558i)13-s + (−1.21 + 0.919i)14-s + (−0.436 + 4.71i)15-s + (0.171 − 0.227i)16-s + (1.05 + 3.98i)17-s + ⋯
L(s)  = 1  + (−1.36 − 0.846i)2-s + (−0.152 + 0.814i)3-s + (0.706 + 1.41i)4-s + (1.46 − 0.136i)5-s + (0.897 − 0.984i)6-s + (0.0916 − 0.236i)7-s + (0.0869 − 0.937i)8-s + (0.292 + 0.113i)9-s + (−2.12 − 1.05i)10-s + (−1.19 + 0.595i)11-s + (−1.26 + 0.359i)12-s + (−0.0143 + 0.154i)13-s + (−0.325 + 0.245i)14-s + (−0.112 + 1.21i)15-s + (0.0429 − 0.0568i)16-s + (0.256 + 0.966i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.974 - 0.222i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ 0.974 - 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.789213 + 0.0889459i\)
\(L(\frac12)\) \(\approx\) \(0.789213 + 0.0889459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.05 - 3.98i)T \)
good2 \( 1 + (1.93 + 1.19i)T + (0.891 + 1.79i)T^{2} \)
3 \( 1 + (0.263 - 1.41i)T + (-2.79 - 1.08i)T^{2} \)
5 \( 1 + (-3.28 + 0.304i)T + (4.91 - 0.918i)T^{2} \)
7 \( 1 + (-0.242 + 0.625i)T + (-5.17 - 4.71i)T^{2} \)
11 \( 1 + (3.96 - 1.97i)T + (6.62 - 8.77i)T^{2} \)
13 \( 1 + (0.0517 - 0.558i)T + (-12.7 - 2.38i)T^{2} \)
19 \( 1 + (-2.85 + 1.76i)T + (8.46 - 17.0i)T^{2} \)
23 \( 1 + (-0.0390 + 0.100i)T + (-16.9 - 15.4i)T^{2} \)
29 \( 1 + (-2.07 + 1.03i)T + (17.4 - 23.1i)T^{2} \)
31 \( 1 + (-4.72 - 0.438i)T + (30.4 + 5.69i)T^{2} \)
37 \( 1 + (-5.29 - 1.50i)T + (31.4 + 19.4i)T^{2} \)
41 \( 1 + (-0.482 + 2.57i)T + (-38.2 - 14.8i)T^{2} \)
43 \( 1 + (-2.03 - 2.69i)T + (-11.7 + 41.3i)T^{2} \)
47 \( 1 + (6.92 - 2.68i)T + (34.7 - 31.6i)T^{2} \)
53 \( 1 + (-0.673 - 0.260i)T + (39.1 + 35.7i)T^{2} \)
59 \( 1 + (-8.49 + 7.74i)T + (5.44 - 58.7i)T^{2} \)
61 \( 1 + (-0.416 + 0.456i)T + (-5.62 - 60.7i)T^{2} \)
67 \( 1 + (-1.52 + 0.943i)T + (29.8 - 59.9i)T^{2} \)
71 \( 1 + (3.78 - 9.76i)T + (-52.4 - 47.8i)T^{2} \)
73 \( 1 + (13.0 + 9.86i)T + (19.9 + 70.2i)T^{2} \)
79 \( 1 + (6.49 + 10.4i)T + (-35.2 + 70.7i)T^{2} \)
83 \( 1 + (17.2 - 3.22i)T + (77.3 - 29.9i)T^{2} \)
89 \( 1 + (0.596 + 6.43i)T + (-87.4 + 16.3i)T^{2} \)
97 \( 1 + (2.14 - 5.53i)T + (-71.6 - 65.3i)T^{2} \)
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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

−11.32689459728340229581754526384, −10.40447175173333838255798746563, −10.02861532446614682766442576240, −9.494854049237369938705054637296, −8.393330199681178436888307397291, −7.31625089119502875736557242237, −5.74806199462459265143090840988, −4.63558904066712387975700061705, −2.77197558820454862689836271206, −1.57692747082740778426546747439, 1.07012850689366620700662957768, 2.52127657400966765848774565017, 5.40101858802446945367963590912, 6.08078817727957569563691172149, 7.04493149144012854720736139260, 7.84364560736078501148733083075, 8.827989482323919919310544484222, 9.920796139400214968438598633338, 10.20668140871157807685071095550, 11.59955699910699189515827945637

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