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Properties

Label 2-605-55.4-c1-0-33
Degree $2$
Conductor $605$
Sign $0.530 + 0.847i$
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  + (−0.780 + 1.07i)2-s + (0.0729 + 0.224i)4-s + (1.80 − 1.31i)5-s + (−4.08 + 1.32i)7-s + (−2.82 − 0.917i)8-s + (−2.42 − 1.76i)9-s + 2.96i·10-s + (4.08 − 5.62i)13-s + (1.76 − 5.42i)14-s + (2.80 − 2.04i)16-s + (−0.964 − 1.32i)17-s + (3.78 − 1.23i)18-s + (0.427 + 0.310i)20-s + (1.54 − 4.75i)25-s + (2.85 + 8.78i)26-s + ⋯
L(s)  = 1  + (−0.552 + 0.759i)2-s + (0.0364 + 0.112i)4-s + (0.809 − 0.587i)5-s + (−1.54 + 0.501i)7-s + (−0.998 − 0.324i)8-s + (−0.809 − 0.587i)9-s + 0.939i·10-s + (1.13 − 1.56i)13-s + (0.471 − 1.45i)14-s + (0.702 − 0.510i)16-s + (−0.234 − 0.322i)17-s + (0.893 − 0.290i)18-s + (0.0954 + 0.0693i)20-s + (0.309 − 0.951i)25-s + (0.559 + 1.72i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540301 - 0.299395i\)
\(L(\frac12)\) \(\approx\) \(0.540301 - 0.299395i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.80 + 1.31i)T \)
11 \( 1 \)
good2 \( 1 + (0.780 - 1.07i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (4.08 - 1.32i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.08 + 5.62i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.964 + 1.32i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (7.23 + 5.25i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.01iT - 43T^{2} \)
47 \( 1 + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.23 + 3.80i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (6.47 - 4.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-11.6 + 3.79i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.7 + 14.7i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + (-29.9 - 92.2i)T^{2} \)
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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

−10.12558871775500136958644897651, −9.320041711275944426842096563495, −8.855535817401676192345761731762, −8.049580126614469584122362636773, −6.78399674001508970041661187168, −5.92054766013971137041939159279, −5.68140515755173375364077971541, −3.54834532069450969272537438877, −2.77655104633732605539465926046, −0.39091123239729078801167270786, 1.66105520693562110309394939700, 2.78369092328816747767327329073, 3.74256474515660989607889125062, 5.58512121149001702726533431566, 6.35138158095400648417639140855, 6.95618306560159388789779663233, 8.663521655748998809422879067387, 9.266735938125794405100509554177, 9.960316485996009685210175616722, 10.84350088147157236390713720462

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