Properties

Label 2-17e2-289.13-c1-0-6
Degree $2$
Conductor $289$
Sign $-0.393 - 0.919i$
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  + (1.97 − 0.562i)2-s + (−1.71 + 2.06i)3-s + (1.88 − 1.16i)4-s + (−2.31 − 0.107i)5-s + (−2.22 + 5.04i)6-s + (−2.58 + 3.76i)7-s + (0.305 − 0.335i)8-s + (−0.771 − 4.12i)9-s + (−4.64 + 1.09i)10-s + (0.533 − 2.26i)11-s + (−0.823 + 5.90i)12-s + (4.29 + 3.91i)13-s + (−2.98 + 8.89i)14-s + (4.19 − 4.60i)15-s + (−1.56 + 3.14i)16-s + (3.31 − 2.45i)17-s + ⋯
L(s)  = 1  + (1.39 − 0.397i)2-s + (−0.989 + 1.19i)3-s + (0.944 − 0.584i)4-s + (−1.03 − 0.0479i)5-s + (−0.908 + 2.05i)6-s + (−0.975 + 1.42i)7-s + (0.108 − 0.118i)8-s + (−0.257 − 1.37i)9-s + (−1.46 + 0.345i)10-s + (0.160 − 0.684i)11-s + (−0.237 + 1.70i)12-s + (1.19 + 1.08i)13-s + (−0.796 + 2.37i)14-s + (1.08 − 1.18i)15-s + (−0.391 + 0.785i)16-s + (0.803 − 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.393 - 0.919i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ -0.393 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.697829 + 1.05797i\)
\(L(\frac12)\) \(\approx\) \(0.697829 + 1.05797i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.31 + 2.45i)T \)
good2 \( 1 + (-1.97 + 0.562i)T + (1.70 - 1.05i)T^{2} \)
3 \( 1 + (1.71 - 2.06i)T + (-0.551 - 2.94i)T^{2} \)
5 \( 1 + (2.31 + 0.107i)T + (4.97 + 0.461i)T^{2} \)
7 \( 1 + (2.58 - 3.76i)T + (-2.52 - 6.52i)T^{2} \)
11 \( 1 + (-0.533 + 2.26i)T + (-9.84 - 4.90i)T^{2} \)
13 \( 1 + (-4.29 - 3.91i)T + (1.19 + 12.9i)T^{2} \)
19 \( 1 + (0.0931 + 0.0265i)T + (16.1 + 10.0i)T^{2} \)
23 \( 1 + (-1.02 + 1.49i)T + (-8.30 - 21.4i)T^{2} \)
29 \( 1 + (0.516 + 0.121i)T + (25.9 + 12.9i)T^{2} \)
31 \( 1 + (-0.465 - 10.0i)T + (-30.8 + 2.86i)T^{2} \)
37 \( 1 + (-8.00 + 1.11i)T + (35.5 - 10.1i)T^{2} \)
41 \( 1 + (-0.0723 - 0.0601i)T + (7.53 + 40.3i)T^{2} \)
43 \( 1 + (7.73 - 3.85i)T + (25.9 - 34.3i)T^{2} \)
47 \( 1 + (-8.30 - 1.55i)T + (43.8 + 16.9i)T^{2} \)
53 \( 1 + (1.00 + 5.39i)T + (-49.4 + 19.1i)T^{2} \)
59 \( 1 + (4.63 - 11.9i)T + (-43.6 - 39.7i)T^{2} \)
61 \( 1 + (-1.91 + 4.32i)T + (-41.0 - 45.0i)T^{2} \)
67 \( 1 + (-2.73 + 9.60i)T + (-56.9 - 35.2i)T^{2} \)
71 \( 1 + (-2.33 - 1.60i)T + (25.6 + 66.2i)T^{2} \)
73 \( 1 + (3.23 - 1.08i)T + (58.2 - 43.9i)T^{2} \)
79 \( 1 + (1.58 + 2.84i)T + (-41.5 + 67.1i)T^{2} \)
83 \( 1 + (-2.23 - 0.207i)T + (81.5 + 15.2i)T^{2} \)
89 \( 1 + (2.66 - 2.43i)T + (8.21 - 88.6i)T^{2} \)
97 \( 1 + (-13.5 - 9.27i)T + (35.0 + 90.4i)T^{2} \)
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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

−11.83613787957361924572020001587, −11.60446517243481799673614857342, −10.72917393481822412507428404554, −9.383794074250033408379505357141, −8.584910250814188016696376058940, −6.46095125534115852978044344768, −5.81203496066007751482008912872, −4.87556465760684990454999888234, −3.84330371838581754789265122622, −3.11895148010651159434228553243, 0.70193268104646564397179606368, 3.45629101419323866876562457073, 4.19733971342551238833410059370, 5.72060519369849049336813135376, 6.39220359855330318391817748216, 7.32223886314242237960972173934, 7.81672184592818950431326626084, 9.966601955901477899244801957147, 11.07966282977083874340516415827, 11.84211610564028147339328126578

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