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Properties

Label 2-435-87.17-c1-0-21
Degree $2$
Conductor $435$
Sign $-0.468 - 0.883i$
Analytic cond. $3.47349$
Root an. cond. $1.86373$
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  + (1.82 + 1.82i)2-s + (1.68 + 0.389i)3-s + 4.67i·4-s − 5-s + (2.37 + 3.79i)6-s + 1.47·7-s + (−4.89 + 4.89i)8-s + (2.69 + 1.31i)9-s + (−1.82 − 1.82i)10-s + (−4.06 − 4.06i)11-s + (−1.82 + 7.89i)12-s − 3.06i·13-s + (2.69 + 2.69i)14-s + (−1.68 − 0.389i)15-s − 8.52·16-s + (−2.44 − 2.44i)17-s + ⋯
L(s)  = 1  + (1.29 + 1.29i)2-s + (0.974 + 0.224i)3-s + 2.33i·4-s − 0.447·5-s + (0.968 + 1.54i)6-s + 0.558·7-s + (−1.72 + 1.72i)8-s + (0.898 + 0.438i)9-s + (−0.577 − 0.577i)10-s + (−1.22 − 1.22i)11-s + (−0.525 + 2.27i)12-s − 0.851i·13-s + (0.721 + 0.721i)14-s + (−0.435 − 0.100i)15-s − 2.13·16-s + (−0.591 − 0.591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.468 - 0.883i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ -0.468 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71014 + 2.84371i\)
\(L(\frac12)\) \(\approx\) \(1.71014 + 2.84371i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.389i)T \)
5 \( 1 + T \)
29 \( 1 + (3.32 - 4.23i)T \)
good2 \( 1 + (-1.82 - 1.82i)T + 2iT^{2} \)
7 \( 1 - 1.47T + 7T^{2} \)
11 \( 1 + (4.06 + 4.06i)T + 11iT^{2} \)
13 \( 1 + 3.06iT - 13T^{2} \)
17 \( 1 + (2.44 + 2.44i)T + 17iT^{2} \)
19 \( 1 + (-5.06 + 5.06i)T - 19iT^{2} \)
23 \( 1 - 4.53iT - 23T^{2} \)
31 \( 1 + (6.46 - 6.46i)T - 31iT^{2} \)
37 \( 1 + (-2.96 - 2.96i)T + 37iT^{2} \)
41 \( 1 + (1.91 - 1.91i)T - 41iT^{2} \)
43 \( 1 + (1.94 - 1.94i)T - 43iT^{2} \)
47 \( 1 + (-5.96 + 5.96i)T - 47iT^{2} \)
53 \( 1 + 4.16iT - 53T^{2} \)
59 \( 1 - 4.55iT - 59T^{2} \)
61 \( 1 + (-7.54 + 7.54i)T - 61iT^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 + 3.37T + 71T^{2} \)
73 \( 1 + (1.11 + 1.11i)T + 73iT^{2} \)
79 \( 1 + (-3.77 + 3.77i)T - 79iT^{2} \)
83 \( 1 - 6.93iT - 83T^{2} \)
89 \( 1 + (3.53 + 3.53i)T + 89iT^{2} \)
97 \( 1 + (2.64 + 2.64i)T + 97iT^{2} \)
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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

−11.60150170726351671734200341229, −10.75364185405734774980696041703, −9.153230825465803919687930153267, −8.256411527420293429920216001413, −7.68170230581880874421229104731, −6.95818517184766538037702462129, −5.31416301098267916719810645778, −5.01220403438930688937635364901, −3.51775307224875539932665115475, −2.93907100981855761338936670944, 1.77188944297205765178445739522, 2.53065645701355855520323999870, 3.92328893137295362272722561599, 4.46015077330480502532895155994, 5.67370090125306499881840023556, 7.18210421777543402680850560391, 8.062982830855016150535440320900, 9.412182818629265572597778602787, 10.14143235139102999505463568530, 11.04796649956658864862512700765

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