Properties

Label 2-35-7.6-c2-0-0
Degree $2$
Conductor $35$
Sign $-i$
Analytic cond. $0.953680$
Root an. cond. $0.976565$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4.47i·3-s − 3·4-s + 2.23i·5-s − 4.47i·6-s + 7·7-s + 7·8-s − 11.0·9-s − 2.23i·10-s + 2·11-s − 13.4i·12-s − 13.4i·13-s − 7·14-s − 10.0·15-s + 5·16-s + 26.8i·17-s + ⋯
L(s)  = 1  − 0.5·2-s + 1.49i·3-s − 0.750·4-s + 0.447i·5-s − 0.745i·6-s + 7-s + 0.875·8-s − 1.22·9-s − 0.223i·10-s + 0.181·11-s − 1.11i·12-s − 1.03i·13-s − 0.5·14-s − 0.666·15-s + 0.312·16-s + 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ -i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.533518 + 0.533518i\)
\(L(\frac12)\) \(\approx\) \(0.533518 + 0.533518i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 2.23iT \)
7 \( 1 - 7T \)
good2 \( 1 + T + 4T^{2} \)
3 \( 1 - 4.47iT - 9T^{2} \)
11 \( 1 - 2T + 121T^{2} \)
13 \( 1 + 13.4iT - 169T^{2} \)
17 \( 1 - 26.8iT - 289T^{2} \)
19 \( 1 + 13.4iT - 361T^{2} \)
23 \( 1 - 26T + 529T^{2} \)
29 \( 1 + 22T + 841T^{2} \)
31 \( 1 + 53.6iT - 961T^{2} \)
37 \( 1 - 14T + 1.36e3T^{2} \)
41 \( 1 + 26.8iT - 1.68e3T^{2} \)
43 \( 1 + 34T + 1.84e3T^{2} \)
47 \( 1 + 26.8iT - 2.20e3T^{2} \)
53 \( 1 + 34T + 2.80e3T^{2} \)
59 \( 1 + 40.2iT - 3.48e3T^{2} \)
61 \( 1 - 93.9iT - 3.72e3T^{2} \)
67 \( 1 - 14T + 4.48e3T^{2} \)
71 \( 1 - 62T + 5.04e3T^{2} \)
73 \( 1 + 53.6iT - 5.32e3T^{2} \)
79 \( 1 - 38T + 6.24e3T^{2} \)
83 \( 1 - 40.2iT - 6.88e3T^{2} \)
89 \( 1 + 26.8iT - 7.92e3T^{2} \)
97 \( 1 + 26.8iT - 9.40e3T^{2} \)
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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

−16.90062779357632364130977895505, −15.18054868318445563270778672281, −14.78419501369660308067686865692, −13.20739135960693087104765265339, −11.08651718063902348434576701937, −10.36326966463145767838290231595, −9.164014712372264624353312867507, −7.995953454681235411202412604988, −5.28862869321859185071245216462, −3.96071990847575989867755262542, 1.37313495336699109588713138389, 4.94681256561585551199079696374, 7.08270241161797109038839913690, 8.192361067925638487782006921128, 9.297702931867797212328224996782, 11.34461490680469755766888114602, 12.47399735256140568264933170236, 13.66427625720846890808078326732, 14.34737609418793252514283430911, 16.54163838919304782470716598261

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