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Properties

Label 2-605-55.14-c1-0-13
Degree $2$
Conductor $605$
Sign $-0.0272 - 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.972 + 1.33i)2-s + (−1.87 + 0.610i)3-s + (−0.227 + 0.701i)4-s + (−1.51 − 1.64i)5-s + (−2.64 − 1.92i)6-s + (2.13 + 0.693i)7-s + (1.98 − 0.645i)8-s + (0.727 − 0.528i)9-s + (0.718 − 3.62i)10-s − 1.45i·12-s + (2.17 + 2.99i)13-s + (1.14 + 3.52i)14-s + (3.85 + 2.15i)15-s + (3.98 + 2.89i)16-s + (1.30 − 1.79i)17-s + (1.41 + 0.460i)18-s + ⋯
L(s)  = 1  + (0.687 + 0.946i)2-s + (−1.08 + 0.352i)3-s + (−0.113 + 0.350i)4-s + (−0.679 − 0.733i)5-s + (−1.07 − 0.784i)6-s + (0.806 + 0.261i)7-s + (0.702 − 0.228i)8-s + (0.242 − 0.176i)9-s + (0.227 − 1.14i)10-s − 0.420i·12-s + (0.603 + 0.830i)13-s + (0.306 + 0.943i)14-s + (0.995 + 0.556i)15-s + (0.997 + 0.724i)16-s + (0.317 − 0.436i)17-s + (0.333 + 0.108i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0272 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0272 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.0272 - 0.999i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.0272 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06108 + 1.09041i\)
\(L(\frac12)\) \(\approx\) \(1.06108 + 1.09041i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.51 + 1.64i)T \)
11 \( 1 \)
good2 \( 1 + (-0.972 - 1.33i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (1.87 - 0.610i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-2.13 - 0.693i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.17 - 2.99i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.30 + 1.79i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.63 - 5.02i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.85iT - 23T^{2} \)
29 \( 1 + (-0.0582 + 0.179i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.555 - 0.403i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.46 - 0.801i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.44 + 7.52i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.41iT - 43T^{2} \)
47 \( 1 + (-11.4 + 3.71i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.43 - 10.2i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.106 + 0.326i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.40 - 1.01i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 0.650iT - 67T^{2} \)
71 \( 1 + (3.75 + 2.73i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.42 + 2.73i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.85 + 4.25i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.87 + 2.57i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 + (1.33 + 1.83i)T + (-29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.11443943923114551755224819810, −10.17543828402502845234936752298, −8.937634961785010054826788543665, −7.966807525331198892882004536145, −7.21256476902489539457828728613, −5.95234909043132360642869319101, −5.45705262841798989707863348366, −4.65007659226255342221133826913, −3.89934851880068493106779394696, −1.35902316281157206313558666167, 0.943675123753452921043135987839, 2.62055672568098733265317648033, 3.70634355518937554393029857134, 4.71426742304981025359110991587, 5.64192297646447902895342227338, 6.78565158189343820401055328207, 7.63230754253007257033432813628, 8.506567169147114188663561487891, 10.26946723795693337931158913123, 10.87095313989855543342090044808

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