Properties

Label 2-605-55.43-c1-0-16
Degree $2$
Conductor $605$
Sign $-0.841 - 0.540i$
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  + (1.64 + 1.64i)2-s + (1.43 + 1.43i)3-s + 3.44i·4-s + (1.13 + 1.92i)5-s + 4.75i·6-s + (−3.25 − 3.25i)7-s + (−2.37 + 2.37i)8-s + 1.14i·9-s + (−1.30 + 5.05i)10-s + (−4.95 + 4.95i)12-s + (−1.73 + 1.73i)13-s − 10.7i·14-s + (−1.13 + 4.40i)15-s − 0.961·16-s + (1.32 + 1.32i)17-s + (−1.89 + 1.89i)18-s + ⋯
L(s)  = 1  + (1.16 + 1.16i)2-s + (0.831 + 0.831i)3-s + 1.72i·4-s + (0.507 + 0.861i)5-s + 1.93i·6-s + (−1.23 − 1.23i)7-s + (−0.840 + 0.840i)8-s + 0.382i·9-s + (−0.412 + 1.59i)10-s + (−1.43 + 1.43i)12-s + (−0.480 + 0.480i)13-s − 2.87i·14-s + (−0.294 + 1.13i)15-s − 0.240·16-s + (0.322 + 0.322i)17-s + (−0.445 + 0.445i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.841 - 0.540i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939440 + 3.20368i\)
\(L(\frac12)\) \(\approx\) \(0.939440 + 3.20368i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.13 - 1.92i)T \)
11 \( 1 \)
good2 \( 1 + (-1.64 - 1.64i)T + 2iT^{2} \)
3 \( 1 + (-1.43 - 1.43i)T + 3iT^{2} \)
7 \( 1 + (3.25 + 3.25i)T + 7iT^{2} \)
13 \( 1 + (1.73 - 1.73i)T - 13iT^{2} \)
17 \( 1 + (-1.32 - 1.32i)T + 17iT^{2} \)
19 \( 1 - 1.71T + 19T^{2} \)
23 \( 1 + (-0.313 - 0.313i)T + 23iT^{2} \)
29 \( 1 - 2.83T + 29T^{2} \)
31 \( 1 - 2.34T + 31T^{2} \)
37 \( 1 + (-1.29 + 1.29i)T - 37iT^{2} \)
41 \( 1 + 7.57iT - 41T^{2} \)
43 \( 1 + (-4.63 + 4.63i)T - 43iT^{2} \)
47 \( 1 + (-1.55 + 1.55i)T - 47iT^{2} \)
53 \( 1 + (9.45 + 9.45i)T + 53iT^{2} \)
59 \( 1 + 4.14iT - 59T^{2} \)
61 \( 1 + 2.45iT - 61T^{2} \)
67 \( 1 + (-8.24 + 8.24i)T - 67iT^{2} \)
71 \( 1 + 9.33T + 71T^{2} \)
73 \( 1 + (9.98 - 9.98i)T - 73iT^{2} \)
79 \( 1 + 2.12T + 79T^{2} \)
83 \( 1 + (10.5 - 10.5i)T - 83iT^{2} \)
89 \( 1 - 4.75iT - 89T^{2} \)
97 \( 1 + (9.77 - 9.77i)T - 97iT^{2} \)
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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

−10.73679690278479964282628366476, −9.959264353606630957906042008851, −9.431174223656227110216237444240, −8.071809540056816549530743612937, −7.01869854596570614248237499010, −6.67125913648234531643629344740, −5.59155620357481277306162983143, −4.28334386343161059281130788523, −3.63695378966488510533794348912, −2.88553261189461659511390546496, 1.37083633642121902815044215265, 2.71029119744681962481246568681, 2.91219611671402194997920165233, 4.57257073366137865127607899768, 5.55259409432498595882425490662, 6.27764206468945071196330439922, 7.72880958472696191566627770323, 8.777092535438074705162643770535, 9.552261222113207691993414213574, 10.23995120953294771196873625517

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