Properties

Label 2-17e2-289.106-c1-0-17
Degree $2$
Conductor $289$
Sign $0.328 + 0.944i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.329 − 0.849i)2-s + (−0.299 − 2.14i)3-s + (0.864 + 0.788i)4-s + (0.894 + 2.66i)5-s + (−1.92 − 0.452i)6-s + (0.708 − 1.27i)7-s + (2.58 − 1.28i)8-s + (−1.63 + 0.466i)9-s + (2.56 + 0.118i)10-s + (−0.131 − 2.83i)11-s + (1.43 − 2.09i)12-s + (0.606 + 1.21i)13-s + (−0.847 − 1.02i)14-s + (5.46 − 2.72i)15-s + (−0.0266 − 0.287i)16-s + (−0.991 + 4.00i)17-s + ⋯
L(s)  = 1  + (0.232 − 0.600i)2-s + (−0.172 − 1.24i)3-s + (0.432 + 0.394i)4-s + (0.399 + 1.19i)5-s + (−0.785 − 0.184i)6-s + (0.267 − 0.480i)7-s + (0.913 − 0.455i)8-s + (−0.546 + 0.155i)9-s + (0.809 + 0.0374i)10-s + (−0.0395 − 0.855i)11-s + (0.414 − 0.604i)12-s + (0.168 + 0.338i)13-s + (−0.226 − 0.272i)14-s + (1.41 − 0.702i)15-s + (−0.00666 − 0.0719i)16-s + (−0.240 + 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ 0.328 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39605 - 0.992227i\)
\(L(\frac12)\) \(\approx\) \(1.39605 - 0.992227i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.991 - 4.00i)T \)
good2 \( 1 + (-0.329 + 0.849i)T + (-1.47 - 1.34i)T^{2} \)
3 \( 1 + (0.299 + 2.14i)T + (-2.88 + 0.820i)T^{2} \)
5 \( 1 + (-0.894 - 2.66i)T + (-3.99 + 3.01i)T^{2} \)
7 \( 1 + (-0.708 + 1.27i)T + (-3.68 - 5.95i)T^{2} \)
11 \( 1 + (0.131 + 2.83i)T + (-10.9 + 1.01i)T^{2} \)
13 \( 1 + (-0.606 - 1.21i)T + (-7.83 + 10.3i)T^{2} \)
19 \( 1 + (2.01 + 5.18i)T + (-14.0 + 12.8i)T^{2} \)
23 \( 1 + (2.22 - 3.99i)T + (-12.1 - 19.5i)T^{2} \)
29 \( 1 + (-2.62 + 0.121i)T + (28.8 - 2.67i)T^{2} \)
31 \( 1 + (7.46 + 2.50i)T + (24.7 + 18.6i)T^{2} \)
37 \( 1 + (7.74 - 5.30i)T + (13.3 - 34.5i)T^{2} \)
41 \( 1 + (0.892 - 0.124i)T + (39.4 - 11.2i)T^{2} \)
43 \( 1 + (-2.28 - 0.211i)T + (42.2 + 7.90i)T^{2} \)
47 \( 1 + (0.732 - 2.57i)T + (-39.9 - 24.7i)T^{2} \)
53 \( 1 + (-6.27 + 1.78i)T + (45.0 - 27.9i)T^{2} \)
59 \( 1 + (5.23 - 8.44i)T + (-26.2 - 52.8i)T^{2} \)
61 \( 1 + (-12.7 - 3.00i)T + (54.6 + 27.1i)T^{2} \)
67 \( 1 + (4.10 - 1.59i)T + (49.5 - 45.1i)T^{2} \)
71 \( 1 + (-7.65 - 4.26i)T + (37.3 + 60.3i)T^{2} \)
73 \( 1 + (6.37 + 5.29i)T + (13.4 + 71.7i)T^{2} \)
79 \( 1 + (9.25 + 4.08i)T + (53.2 + 58.3i)T^{2} \)
83 \( 1 + (0.991 - 0.748i)T + (22.7 - 79.8i)T^{2} \)
89 \( 1 + (-3.00 + 6.02i)T + (-53.6 - 71.0i)T^{2} \)
97 \( 1 + (16.6 + 9.24i)T + (51.0 + 82.4i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.51476610174222967012280164298, −10.98915657817449814251332121976, −10.22048606642253056367352466939, −8.527607419643268019801974716383, −7.38847460658617648142532354178, −6.85390005829914042068705066902, −5.99800445388978433845052326906, −3.96119033297239196729121589059, −2.72385941688839626405167722570, −1.61703934121203732507647434596, 1.93122103283321967722015944066, 4.14988383996009625851802189475, 5.15769230038381073069186109468, 5.50885433343069697322388252159, 6.96168041629669047678696078212, 8.297186382001940187502523282421, 9.235821806886559428185019620746, 10.10590686072340216566165683240, 10.79975161778602119396913411200, 12.03015384152640186647004508705

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