↑

Properties

Label 2-731-731.208-c1-0-33
Degree $2$
Conductor $731$
Sign $-0.353 + 0.935i$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
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  − 1.62i·2-s + (−0.958 − 0.256i)3-s − 0.637·4-s + (3.71 + 0.994i)5-s + (−0.417 + 1.55i)6-s + (−0.920 + 0.246i)7-s − 2.21i·8-s + (−1.74 − 1.00i)9-s + (1.61 − 6.02i)10-s + (1.86 + 1.86i)11-s + (0.611 + 0.163i)12-s + (−0.714 + 1.23i)13-s + (0.400 + 1.49i)14-s + (−3.30 − 1.90i)15-s − 4.86·16-s + (1.60 − 3.79i)17-s + ⋯
L(s)  = 1  − 1.14i·2-s + (−0.553 − 0.148i)3-s − 0.318·4-s + (1.66 + 0.444i)5-s + (−0.170 + 0.635i)6-s + (−0.348 + 0.0932i)7-s − 0.782i·8-s + (−0.581 − 0.335i)9-s + (0.510 − 1.90i)10-s + (0.561 + 0.561i)11-s + (0.176 + 0.0472i)12-s + (−0.198 + 0.343i)13-s + (0.107 + 0.399i)14-s + (−0.852 − 0.492i)15-s − 1.21·16-s + (0.390 − 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.353 + 0.935i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.353 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975789 - 1.41216i\)
\(L(\frac12)\) \(\approx\) \(0.975789 - 1.41216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.60 + 3.79i)T \)
43 \( 1 + (-4.36 - 4.89i)T \)
good2 \( 1 + 1.62iT - 2T^{2} \)
3 \( 1 + (0.958 + 0.256i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-3.71 - 0.994i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.920 - 0.246i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.86 - 1.86i)T + 11iT^{2} \)
13 \( 1 + (0.714 - 1.23i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (-3.40 + 1.96i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.26 + 4.72i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.32 + 8.66i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-5.90 - 1.58i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-4.18 - 1.12i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (5.76 + 5.76i)T + 41iT^{2} \)
47 \( 1 - 4.05T + 47T^{2} \)
53 \( 1 + (-3.82 + 2.20i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.12iT - 59T^{2} \)
61 \( 1 + (13.2 - 3.55i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.708 - 1.22i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.16 - 8.09i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (2.56 + 9.59i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.84 - 0.494i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (15.2 - 8.81i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.391 + 0.678i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.972 - 0.972i)T - 97iT^{2} \)
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

−10.08526339618183235354498168462, −9.626384237304543094513204726196, −9.013523761027366352123860984796, −7.15760269835163934762581643456, −6.46229774077251540274834414753, −5.78768189254099926617822323631, −4.57522515393406731225318266469, −2.98904905920291649704764617580, −2.38356715323491818862125985548, −1.04098876341739001233255869860, 1.55642926693614669019654507448, 3.05183489488924460403301548602, 4.90340647739274487345560890985, 5.75006613098415555804693053157, 5.90697176145515068048317953307, 6.87675756449562095125415280976, 8.042531527617436898749200403001, 8.858908827024439723696609627629, 9.677996682382970898255125993818, 10.52189478141175745581087261988

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