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Properties

Label 2-605-55.32-c1-0-38
Degree $2$
Conductor $605$
Sign $-0.501 + 0.865i$
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  + (1.79 − 1.79i)2-s + (−1.41 + 1.41i)3-s − 4.42i·4-s + (1.69 − 1.46i)5-s + 5.08i·6-s + (1.27 − 1.27i)7-s + (−4.35 − 4.35i)8-s − 1.01i·9-s + (0.412 − 5.65i)10-s + (6.27 + 6.27i)12-s + (−1.59 − 1.59i)13-s − 4.58i·14-s + (−0.326 + 4.47i)15-s − 6.74·16-s + (2.37 − 2.37i)17-s + (−1.82 − 1.82i)18-s + ⋯
L(s)  = 1  + (1.26 − 1.26i)2-s + (−0.818 + 0.818i)3-s − 2.21i·4-s + (0.756 − 0.653i)5-s + 2.07i·6-s + (0.483 − 0.483i)7-s + (−1.53 − 1.53i)8-s − 0.339i·9-s + (0.130 − 1.78i)10-s + (1.81 + 1.81i)12-s + (−0.442 − 0.442i)13-s − 1.22i·14-s + (−0.0842 + 1.15i)15-s − 1.68·16-s + (0.576 − 0.576i)17-s + (−0.430 − 0.430i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.501 + 0.865i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.501 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23555 - 2.14491i\)
\(L(\frac12)\) \(\approx\) \(1.23555 - 2.14491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.69 + 1.46i)T \)
11 \( 1 \)
good2 \( 1 + (-1.79 + 1.79i)T - 2iT^{2} \)
3 \( 1 + (1.41 - 1.41i)T - 3iT^{2} \)
7 \( 1 + (-1.27 + 1.27i)T - 7iT^{2} \)
13 \( 1 + (1.59 + 1.59i)T + 13iT^{2} \)
17 \( 1 + (-2.37 + 2.37i)T - 17iT^{2} \)
19 \( 1 - 1.30T + 19T^{2} \)
23 \( 1 + (3.16 - 3.16i)T - 23iT^{2} \)
29 \( 1 - 2.93T + 29T^{2} \)
31 \( 1 + 4.01T + 31T^{2} \)
37 \( 1 + (2.26 + 2.26i)T + 37iT^{2} \)
41 \( 1 + 1.30iT - 41T^{2} \)
43 \( 1 + (-4.55 - 4.55i)T + 43iT^{2} \)
47 \( 1 + (-5.46 - 5.46i)T + 47iT^{2} \)
53 \( 1 + (-4.50 + 4.50i)T - 53iT^{2} \)
59 \( 1 - 12.3iT - 59T^{2} \)
61 \( 1 - 9.31iT - 61T^{2} \)
67 \( 1 + (2.46 + 2.46i)T + 67iT^{2} \)
71 \( 1 + 6.77T + 71T^{2} \)
73 \( 1 + (-6.67 - 6.67i)T + 73iT^{2} \)
79 \( 1 - 1.14T + 79T^{2} \)
83 \( 1 + (-7.39 - 7.39i)T + 83iT^{2} \)
89 \( 1 - 3.85iT - 89T^{2} \)
97 \( 1 + (0.550 + 0.550i)T + 97iT^{2} \)
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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

−10.47308143723222031442725076517, −10.03850989022170532761258961072, −9.217511773048210580847628490355, −7.61097140970552453079888282660, −5.92959464919577924223424604735, −5.39338655130712196801112048185, −4.71843128783914305900174050336, −3.92594520879570385896784433299, −2.50455112390673274991996669797, −1.11155549253365341082348004484, 2.07336450671756479504494651237, 3.54593634912676991996107056341, 4.93583361530467638612044923531, 5.70239937879882407620680531478, 6.28845493578070223973843293622, 7.00567324042692751783810818033, 7.73953841257270583749934124678, 8.858843236365894226136785771982, 10.20141261542148192295948376652, 11.36014943714965729354033083968

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