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Properties

Label 2-605-55.39-c0-0-0
Degree $2$
Conductor $605$
Sign $0.964 - 0.265i$
Analytic cond. $0.301934$
Root an. cond. $0.549485$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)20-s + (−0.809 − 0.587i)25-s + (−0.618 − 1.90i)31-s + (0.809 + 0.587i)36-s − 0.999·45-s + (−0.309 + 0.951i)49-s + (−1.61 + 1.17i)59-s + (−0.309 − 0.951i)64-s + (0.618 − 1.90i)71-s + (0.809 + 0.587i)80-s + (−0.809 + 0.587i)81-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)20-s + (−0.809 − 0.587i)25-s + (−0.618 − 1.90i)31-s + (0.809 + 0.587i)36-s − 0.999·45-s + (−0.309 + 0.951i)49-s + (−1.61 + 1.17i)59-s + (−0.309 − 0.951i)64-s + (0.618 − 1.90i)71-s + (0.809 + 0.587i)80-s + (−0.809 + 0.587i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(0.301934\)
Root analytic conductor: \(0.549485\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :0),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.043642640\)
\(L(\frac12)\) \(\approx\) \(1.043642640\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−10.92578666281152917750732850494, −10.25881851021832487604136661434, −9.412595617578956705496853738427, −7.88473133727794176386653643295, −7.40935903978838611697412720426, −6.43228875898252587231759117997, −5.61084689896311571705213913272, −4.33391648754884933245420015259, −2.93444875780730462281345594157, −1.93699773126312220587693635294, 1.54717661651294741362798882938, 3.19098838552227910320030886240, 4.09347618665693302645786346977, 5.31204243810694259474995433068, 6.49138027950802718578027521651, 7.24984748089888624073211011470, 8.250934098712135693965273501209, 8.938020424124444982098554098999, 9.908577561828789285357601704170, 11.00714249449202219913876527928

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