↑

Properties

Label 2-17e2-17.13-c1-0-12
Degree $2$
Conductor $289$
Sign $-0.281 + 0.959i$
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  − 0.347i·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

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.281 + 0.959i$
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: \((2,\ 289,\ (\ :1/2),\ -0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.673403 - 0.899430i\)
\(L(\frac12)\) \(\approx\) \(0.673403 - 0.899430i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.55 - 6.55i)T + 97iT^{2} \)
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

−11.91008540250634824412139898140, −10.89859017269838015711657497315, −9.562763510593988441023097776771, −8.755140208955856296285469467103, −7.49332737007406336405580120702, −6.56661811505191148995638573097, −5.72733609883198216892458760407, −4.08221062692988132998425629588, −2.82625177916006820352506347964, −0.889036812151915290827017012863, 2.30121812367246822095942995020, 3.75004181109680975699121574751, 4.97137441392615392227077646193, 6.43480966948599474928284086212, 7.23956515818645196945707763667, 7.73232053664915910077085183656, 9.547397117190731965957054833857, 10.33170522438701456107874605661, 11.17931395405927382556304804914, 11.85386756676264146303646649406

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