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Properties

Label 2-605-55.32-c1-0-7
Degree $2$
Conductor $605$
Sign $-0.878 - 0.478i$
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  + (−0.436 + 0.436i)2-s + (1.06 − 1.06i)3-s + 1.61i·4-s + (−1.07 + 1.95i)5-s + 0.928i·6-s + (−1.08 + 1.08i)7-s + (−1.58 − 1.58i)8-s + 0.741i·9-s + (−0.384 − 1.32i)10-s + (1.71 + 1.71i)12-s + (−0.147 − 0.147i)13-s − 0.950i·14-s + (0.935 + 3.22i)15-s − 1.85·16-s + (2.69 − 2.69i)17-s + (−0.324 − 0.324i)18-s + ⋯
L(s)  = 1  + (−0.308 + 0.308i)2-s + (0.613 − 0.613i)3-s + 0.809i·4-s + (−0.482 + 0.876i)5-s + 0.378i·6-s + (−0.411 + 0.411i)7-s + (−0.558 − 0.558i)8-s + 0.247i·9-s + (−0.121 − 0.419i)10-s + (0.496 + 0.496i)12-s + (−0.0407 − 0.0407i)13-s − 0.254i·14-s + (0.241 + 0.833i)15-s − 0.464·16-s + (0.653 − 0.653i)17-s + (−0.0763 − 0.0763i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.878 - 0.478i$
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: \((2,\ 605,\ (\ :1/2),\ -0.878 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209467 + 0.821940i\)
\(L(\frac12)\) \(\approx\) \(0.209467 + 0.821940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.07 - 1.95i)T \)
11 \( 1 \)
good2 \( 1 + (0.436 - 0.436i)T - 2iT^{2} \)
3 \( 1 + (-1.06 + 1.06i)T - 3iT^{2} \)
7 \( 1 + (1.08 - 1.08i)T - 7iT^{2} \)
13 \( 1 + (0.147 + 0.147i)T + 13iT^{2} \)
17 \( 1 + (-2.69 + 2.69i)T - 17iT^{2} \)
19 \( 1 + 6.52T + 19T^{2} \)
23 \( 1 + (4.95 - 4.95i)T - 23iT^{2} \)
29 \( 1 + 9.03T + 29T^{2} \)
31 \( 1 - 6.06T + 31T^{2} \)
37 \( 1 + (0.916 + 0.916i)T + 37iT^{2} \)
41 \( 1 - 5.01iT - 41T^{2} \)
43 \( 1 + (-7.33 - 7.33i)T + 43iT^{2} \)
47 \( 1 + (-0.236 - 0.236i)T + 47iT^{2} \)
53 \( 1 + (-3.74 + 3.74i)T - 53iT^{2} \)
59 \( 1 + 5.21iT - 59T^{2} \)
61 \( 1 - 14.7iT - 61T^{2} \)
67 \( 1 + (1.11 + 1.11i)T + 67iT^{2} \)
71 \( 1 + 1.37T + 71T^{2} \)
73 \( 1 + (-9.27 - 9.27i)T + 73iT^{2} \)
79 \( 1 - 5.52T + 79T^{2} \)
83 \( 1 + (5.84 + 5.84i)T + 83iT^{2} \)
89 \( 1 + 9.01iT - 89T^{2} \)
97 \( 1 + (-5.99 - 5.99i)T + 97iT^{2} \)
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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

−11.12962942288826845105019554453, −9.978602847941193865557696464466, −9.065233735032729834583851674860, −8.064342650444223174396988216315, −7.66988459546456664538677255576, −6.86025284522204658667584893720, −5.92989321492603766904893343816, −4.16870471842861434567335173052, −3.13106158744670581047356842697, −2.30605867525789320326405978964, 0.45961763288195176678091250587, 2.10471081471895121330579325595, 3.72064206598889070349042305231, 4.38581072002553665661553977696, 5.65018432620879083216596084155, 6.58931533102745498847322601974, 8.040728183114750224581077481502, 8.749560857843263768965774629600, 9.419419908118576000418607973822, 10.22383186406940775797309316857

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