Properties

Label 2-17-17.4-c3-0-2
Degree $2$
Conductor $17$
Sign $0.999 + 0.0218i$
Analytic cond. $1.00303$
Root an. cond. $1.00151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.351i·2-s + (2.28 − 2.28i)3-s + 7.87·4-s + (−9.32 + 9.32i)5-s + (0.800 + 0.800i)6-s + (−23.5 − 23.5i)7-s + 5.57i·8-s + 16.5i·9-s + (−3.27 − 3.27i)10-s + (9.63 + 9.63i)11-s + (17.9 − 17.9i)12-s − 5.12·13-s + (8.26 − 8.26i)14-s + 42.5i·15-s + 61.0·16-s + (44.3 − 54.2i)17-s + ⋯
L(s)  = 1  + 0.124i·2-s + (0.439 − 0.439i)3-s + 0.984·4-s + (−0.834 + 0.834i)5-s + (0.0544 + 0.0544i)6-s + (−1.27 − 1.27i)7-s + 0.246i·8-s + 0.614i·9-s + (−0.103 − 0.103i)10-s + (0.264 + 0.264i)11-s + (0.432 − 0.432i)12-s − 0.109·13-s + (0.157 − 0.157i)14-s + 0.732i·15-s + 0.954·16-s + (0.632 − 0.774i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0218i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.999 + 0.0218i$
Analytic conductor: \(1.00303\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :3/2),\ 0.999 + 0.0218i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.14108 - 0.0124600i\)
\(L(\frac12)\) \(\approx\) \(1.14108 - 0.0124600i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-44.3 + 54.2i)T \)
good2 \( 1 - 0.351iT - 8T^{2} \)
3 \( 1 + (-2.28 + 2.28i)T - 27iT^{2} \)
5 \( 1 + (9.32 - 9.32i)T - 125iT^{2} \)
7 \( 1 + (23.5 + 23.5i)T + 343iT^{2} \)
11 \( 1 + (-9.63 - 9.63i)T + 1.33e3iT^{2} \)
13 \( 1 + 5.12T + 2.19e3T^{2} \)
19 \( 1 + 38.4iT - 6.85e3T^{2} \)
23 \( 1 + (-13.2 - 13.2i)T + 1.21e4iT^{2} \)
29 \( 1 + (149. - 149. i)T - 2.43e4iT^{2} \)
31 \( 1 + (-165. + 165. i)T - 2.97e4iT^{2} \)
37 \( 1 + (-1.96 + 1.96i)T - 5.06e4iT^{2} \)
41 \( 1 + (214. + 214. i)T + 6.89e4iT^{2} \)
43 \( 1 - 149. iT - 7.95e4T^{2} \)
47 \( 1 + 366.T + 1.03e5T^{2} \)
53 \( 1 + 499. iT - 1.48e5T^{2} \)
59 \( 1 - 507. iT - 2.05e5T^{2} \)
61 \( 1 + (23.3 + 23.3i)T + 2.26e5iT^{2} \)
67 \( 1 - 442.T + 3.00e5T^{2} \)
71 \( 1 + (336. - 336. i)T - 3.57e5iT^{2} \)
73 \( 1 + (520. - 520. i)T - 3.89e5iT^{2} \)
79 \( 1 + (151. + 151. i)T + 4.93e5iT^{2} \)
83 \( 1 + 1.18e3iT - 5.71e5T^{2} \)
89 \( 1 - 325.T + 7.04e5T^{2} \)
97 \( 1 + (-169. + 169. i)T - 9.12e5iT^{2} \)
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

−18.94661598236584128982643303434, −16.82232628559049666276823238935, −15.89675947344895766489601758415, −14.52731742111301042910083823201, −13.13855181094703508558654024490, −11.44915215809469945688305825528, −10.19228075879783336397582502162, −7.51881575581137225411035006800, −6.90363798133140020169176602221, −3.21084383272816080884314557330, 3.36869997447903686472112275929, 6.20715737268211484634574494125, 8.419785836937399860036460283773, 9.793511341496901751846460700170, 11.90381463012337046776509767745, 12.55365951775387738372626525003, 15.02919414624140641179028646484, 15.74297704337318157948957430948, 16.65505715293710863374442790654, 18.96784474015670650225696139595

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