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Properties

Label 2-17e2-17.13-c1-0-9
Degree $2$
Conductor $289$
Sign $0.993 - 0.115i$
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  − 0.347i·2-s + (0.621 + 0.621i)3-s + 1.87·4-s + (1.65 + 1.65i)5-s + (0.215 − 0.215i)6-s + (1.32 − 1.32i)7-s − 1.34i·8-s − 2.22i·9-s + (0.576 − 0.576i)10-s + (−3.58 + 3.58i)11-s + (1.16 + 1.16i)12-s − 4.71·13-s + (−0.461 − 0.461i)14-s + 2.06i·15-s + 3.29·16-s + ⋯
L(s)  = 1  − 0.245i·2-s + (0.359 + 0.359i)3-s + 0.939·4-s + (0.742 + 0.742i)5-s + (0.0881 − 0.0881i)6-s + (0.502 − 0.502i)7-s − 0.476i·8-s − 0.742i·9-s + (0.182 − 0.182i)10-s + (−1.07 + 1.07i)11-s + (0.337 + 0.337i)12-s − 1.30·13-s + (−0.123 − 0.123i)14-s + 0.532i·15-s + 0.822·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.993 - 0.115i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ 0.993 - 0.115i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85765 + 0.108040i\)
\(L(\frac12)\) \(\approx\) \(1.85765 + 0.108040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + 0.347iT - 2T^{2} \)
3 \( 1 + (-0.621 - 0.621i)T + 3iT^{2} \)
5 \( 1 + (-1.65 - 1.65i)T + 5iT^{2} \)
7 \( 1 + (-1.32 + 1.32i)T - 7iT^{2} \)
11 \( 1 + (3.58 - 3.58i)T - 11iT^{2} \)
13 \( 1 + 4.71T + 13T^{2} \)
19 \( 1 + 0.347iT - 19T^{2} \)
23 \( 1 + (1.25 - 1.25i)T - 23iT^{2} \)
29 \( 1 + (1.57 + 1.57i)T + 29iT^{2} \)
31 \( 1 + (-1.37 - 1.37i)T + 31iT^{2} \)
37 \( 1 + (4.36 + 4.36i)T + 37iT^{2} \)
41 \( 1 + (3.65 - 3.65i)T - 41iT^{2} \)
43 \( 1 + 1.47iT - 43T^{2} \)
47 \( 1 - 8.53T + 47T^{2} \)
53 \( 1 - 10.4iT - 53T^{2} \)
59 \( 1 - 5.00iT - 59T^{2} \)
61 \( 1 + (-0.130 + 0.130i)T - 61iT^{2} \)
67 \( 1 + 2.44T + 67T^{2} \)
71 \( 1 + (7.01 + 7.01i)T + 71iT^{2} \)
73 \( 1 + (7.70 + 7.70i)T + 73iT^{2} \)
79 \( 1 + (-3.13 + 3.13i)T - 79iT^{2} \)
83 \( 1 + 13.5iT - 83T^{2} \)
89 \( 1 - 6.32T + 89T^{2} \)
97 \( 1 + (6.55 + 6.55i)T + 97iT^{2} \)
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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

−11.85554510280896340661121780202, −10.41252824670765703708937492115, −10.34759570719898394922599425127, −9.318652362766941196590078609444, −7.63498909549842704601932422252, −7.11581907244156335436678883302, −5.93482784816783034335056807985, −4.53822253862207448820520952543, −2.98205803462285372685353381582, −2.07923753969527503083449173672, 1.89342529358154101837870604884, 2.75762471061991721058792954617, 5.15671641773779622846779659050, 5.56231200961679371106834565293, 7.01980533216017261119160156114, 8.017540571378965133723695484553, 8.595524710236535081972235477269, 9.981178755454422593048254422848, 10.85096243579771065331863205692, 11.85020377426192721440389639582

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