Properties

Label 2-35-5.4-c1-0-0
Degree $2$
Conductor $35$
Sign $0.447 - 0.894i$
Analytic cond. $0.279476$
Root an. cond. $0.528655$
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 i·7-s + 2·9-s + (2 − 4i)10-s − 3·11-s + 2i·12-s i·13-s + 2·14-s + (−1 + 2i)15-s − 4·16-s + 7i·17-s + 4i·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 − 0.377i·7-s + 0.666·9-s + (0.632 − 1.26i)10-s − 0.904·11-s + 0.577i·12-s − 0.277i·13-s + 0.534·14-s + (−0.258 + 0.516i)15-s − 16-s + 1.69i·17-s + 0.942i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(0.279476\)
Root analytic conductor: \(0.528655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.581647 + 0.359478i\)
\(L(\frac12)\) \(\approx\) \(0.581647 + 0.359478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2 + i)T \)
7 \( 1 + iT \)
good2 \( 1 - 2iT - 2T^{2} \)
3 \( 1 + iT - 3T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 7iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−16.58448451363154192517734140835, −15.70323106477834300119154299758, −14.84551408076473918952506948409, −13.34283107156104634236933164891, −12.39216276691738128396285050555, −10.57199601674620508871050040716, −8.382229318624263597580461235223, −7.69057201102210773213766625761, −6.41057499424661595037771355877, −4.58669105072248681251752024317, 3.02592800233480114290664971806, 4.60140633504418596338695192636, 7.35689965275207777678389254869, 9.316604718406395525588995501970, 10.42068121271810089671381063107, 11.42621865015752539898361587727, 12.36148837795374284937339794164, 13.73908249804327735744761172916, 15.51459514519935734353166136015, 15.98210910707013386448206162217

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