↑

Properties

Label 2-605-55.8-c1-0-19
Degree $2$
Conductor $605$
Sign $0.976 - 0.213i$
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  + (−2.20 − 0.349i)2-s + (0.642 − 1.26i)3-s + (2.85 + 0.927i)4-s + (1.03 + 1.98i)5-s + (−1.85 + 2.55i)6-s + (−1.99 − 1.01i)8-s + (0.587 + 0.809i)9-s + (−1.58 − 4.74i)10-s + (3.00 − 3.00i)12-s + (0.699 − 4.41i)13-s + (3.16 − 0.0240i)15-s + (−0.809 − 0.587i)16-s + (0.699 + 4.41i)17-s + (−1.01 − 1.99i)18-s + (1.95 + 6.01i)19-s + (1.09 + 6.61i)20-s + ⋯
L(s)  = 1  + (−1.56 − 0.247i)2-s + (0.370 − 0.727i)3-s + (1.42 + 0.463i)4-s + (0.460 + 0.887i)5-s + (−0.758 + 1.04i)6-s + (−0.704 − 0.358i)8-s + (0.195 + 0.269i)9-s + (−0.500 − 1.50i)10-s + (0.866 − 0.866i)12-s + (0.194 − 1.22i)13-s + (0.816 − 0.00619i)15-s + (−0.202 − 0.146i)16-s + (0.169 + 1.07i)17-s + (−0.239 − 0.469i)18-s + (0.448 + 1.37i)19-s + (0.245 + 1.47i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.976 - 0.213i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.976 - 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.850938 + 0.0921058i\)
\(L(\frac12)\) \(\approx\) \(0.850938 + 0.0921058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.03 - 1.98i)T \)
11 \( 1 \)
good2 \( 1 + (2.20 + 0.349i)T + (1.90 + 0.618i)T^{2} \)
3 \( 1 + (-0.642 + 1.26i)T + (-1.76 - 2.42i)T^{2} \)
7 \( 1 + (4.11 - 5.66i)T^{2} \)
13 \( 1 + (-0.699 + 4.41i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.699 - 4.41i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-1.95 - 6.01i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + (1.95 - 6.01i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.61 - 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.92 + 3.78i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (-6.01 + 1.95i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-3.78 - 1.92i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (-1.39 - 0.221i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-5.70 - 1.85i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.71 + 5.11i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3 + 3i)T - 67iT^{2} \)
71 \( 1 + (-6.47 - 4.70i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.03 + 3.98i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (-5.11 + 3.71i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.83 + 1.39i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (-1.54 + 9.77i)T + (-92.2 - 29.9i)T^{2} \)
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

−10.52193484547326692265204193828, −9.944617933602880892534617785607, −8.884317468083417465169851089085, −7.910260097401332574692152202552, −7.60853042192064891980904081562, −6.58855861078509133532365428743, −5.55772352560035524155396484864, −3.47350725268629048362186623007, −2.29072889833975897724083996167, −1.35945970256799343899447238368, 0.862393605383814311697078633789, 2.30373758488749419966901414673, 4.06633724128670460701781672151, 5.03999557534854458195631986318, 6.45367198444667538121483386934, 7.27740265588926402714814142325, 8.368619652520770046423763899781, 9.190094927098802646347478956657, 9.431543727578662141763107165010, 10.06947096742271055184224137783

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