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Properties

Label 2-435-5.4-c1-0-2
Degree $2$
Conductor $435$
Sign $-0.447 + 0.894i$
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  + 2i·2-s + i·3-s − 2·4-s + (−2 − i)5-s − 2·6-s + 2i·7-s − 9-s + (2 − 4i)10-s − 3·11-s − 2i·12-s + 4i·13-s − 4·14-s + (1 − 2i)15-s − 4·16-s − 8i·17-s − 2i·18-s + ⋯
L(s)  = 1  + 1.41i·2-s + 0.577i·3-s − 4-s + (−0.894 − 0.447i)5-s − 0.816·6-s + 0.755i·7-s − 0.333·9-s + (0.632 − 1.26i)10-s − 0.904·11-s − 0.577i·12-s + 1.10i·13-s − 1.06·14-s + (0.258 − 0.516i)15-s − 16-s − 1.94i·17-s − 0.471i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.341380 - 0.552364i\)
\(L(\frac12)\) \(\approx\) \(0.341380 - 0.552364i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (2 + i)T \)
29 \( 1 + T \)
good2 \( 1 - 2iT - 2T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 8iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + iT - 23T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 - 9iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 13iT - 97T^{2} \)
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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

−11.62422491778398075646995527520, −11.04128124430732654629150603020, −9.325507651786391090280165130585, −9.042466085810642644859319671068, −7.88580792568129009196890952763, −7.31296420995364703643267009715, −6.09357091609398355238453656500, −5.04750629219084679040435311984, −4.50958719877208497077065798184, −2.79056651523504246191001344162, 0.39523673841443701133943970878, 2.04924634891589312439919523686, 3.35942689347374746697520500704, 4.01667183787248106024227404560, 5.63813254884538974270120332872, 7.06648534952353791320988738791, 7.78564118361934078746050748569, 8.724077517634910441749649657607, 10.28587581686152419049399222484, 10.61062742762949293816870953995

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