↑

Properties

Label 2-605-5.4-c1-0-13
Degree $2$
Conductor $605$
Sign $0.533 - 0.845i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + 0.802i·2-s − 1.76i·3-s + 1.35·4-s + (−1.19 + 1.89i)5-s + 1.41·6-s + 0.592i·7-s + 2.69i·8-s − 0.100·9-s + (−1.51 − 0.958i)10-s − 2.38i·12-s + 1.79i·13-s − 0.475·14-s + (3.32 + 2.10i)15-s + 0.549·16-s + 7.07i·17-s − 0.0804i·18-s + ⋯
L(s)  = 1  + 0.567i·2-s − 1.01i·3-s + 0.677·4-s + (−0.533 + 0.845i)5-s + 0.576·6-s + 0.223i·7-s + 0.952i·8-s − 0.0334·9-s + (−0.479 − 0.302i)10-s − 0.689i·12-s + 0.497i·13-s − 0.127·14-s + (0.859 + 0.542i)15-s + 0.137·16-s + 1.71i·17-s − 0.0189i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.533 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44771 + 0.798191i\)
\(L(\frac12)\) \(\approx\) \(1.44771 + 0.798191i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.19 - 1.89i)T \)
11 \( 1 \)
good2 \( 1 - 0.802iT - 2T^{2} \)
3 \( 1 + 1.76iT - 3T^{2} \)
7 \( 1 - 0.592iT - 7T^{2} \)
13 \( 1 - 1.79iT - 13T^{2} \)
17 \( 1 - 7.07iT - 17T^{2} \)
19 \( 1 - 2.28T + 19T^{2} \)
23 \( 1 + 1.49iT - 23T^{2} \)
29 \( 1 + 3.57T + 29T^{2} \)
31 \( 1 - 6.16T + 31T^{2} \)
37 \( 1 - 7.33iT - 37T^{2} \)
41 \( 1 - 8.41T + 41T^{2} \)
43 \( 1 + 9.51iT - 43T^{2} \)
47 \( 1 - 1.93iT - 47T^{2} \)
53 \( 1 + 2.38iT - 53T^{2} \)
59 \( 1 - 0.0382T + 59T^{2} \)
61 \( 1 + 3.44T + 61T^{2} \)
67 \( 1 - 6.79iT - 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 6.82iT - 73T^{2} \)
79 \( 1 + 4.52T + 79T^{2} \)
83 \( 1 + 5.94iT - 83T^{2} \)
89 \( 1 - 6.21T + 89T^{2} \)
97 \( 1 + 5.37iT - 97T^{2} \)
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

−10.87039355637516590065503254105, −10.13923711184625356560670667921, −8.592860498767956602923437532409, −7.84468290916833917900258989131, −7.18635559189743829980027602732, −6.47040996605752551275569916918, −5.84456231081181249748309767712, −4.15914772415065624147338076571, −2.77545400678973259114096416466, −1.69117107894885008637299050104, 0.992755100969675217022314067553, 2.79788973947135586573960849894, 3.83034431546658986416790755787, 4.69809067709490130401499417449, 5.67313566558388252153921251486, 7.16730244194938495031831958481, 7.79137524237368475215689914287, 9.235431166856514270850474542158, 9.618789305194935726637706972486, 10.59895536486893603925764959640

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