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Properties

Label 2-735-105.59-c1-0-1
Degree $2$
Conductor $735$
Sign $0.379 + 0.925i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
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  + (1.36 + 2.36i)2-s + (−1.5 − 0.866i)3-s + (−2.72 + 4.71i)4-s + (−1.93 + 1.11i)5-s − 4.72i·6-s − 9.40·8-s + (1.5 + 2.59i)9-s + (−5.28 − 3.05i)10-s + (8.16 − 4.71i)12-s + 3.87·15-s + (−7.38 − 12.7i)16-s + (−1.50 − 0.868i)17-s + (−4.09 + 7.08i)18-s + (−0.633 + 0.365i)19-s − 12.1i·20-s + ⋯
L(s)  = 1  + (0.964 + 1.67i)2-s + (−0.866 − 0.499i)3-s + (−1.36 + 2.35i)4-s + (−0.866 + 0.499i)5-s − 1.92i·6-s − 3.32·8-s + (0.5 + 0.866i)9-s + (−1.67 − 0.964i)10-s + (2.35 − 1.36i)12-s + 1.00·15-s + (−1.84 − 3.19i)16-s + (−0.364 − 0.210i)17-s + (−0.964 + 1.67i)18-s + (−0.145 + 0.0838i)19-s − 2.72i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.379 + 0.925i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.379 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278636 - 0.186919i\)
\(L(\frac12)\) \(\approx\) \(0.278636 - 0.186919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 \)
good2 \( 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (1.50 + 0.868i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.633 - 0.365i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.31 + 5.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-3.54 - 2.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (11.8 - 6.83i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.25 - 12.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.7 - 6.77i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.39 + 14.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.1iT - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−11.42540002687409057169397383860, −10.36323941919505449438083017669, −8.844407678035175978787183421973, −7.926001553289077411904792843126, −7.43610335996495044186470683685, −6.46627550062594913351448772119, −6.13265413770993834575311731207, −4.80127392638770088870133528630, −4.31885424542533392178548343095, −2.96228899726449226571025117009, 0.14290459423679411181904900616, 1.58491331983702887863941245395, 3.29815486009201404755093691168, 4.04677631204267386943109816537, 4.80892248539293258061056788169, 5.54322369718546455992688227308, 6.61216688123674386066355707851, 8.257504797086117390326218640680, 9.410556183388722186928516084835, 9.949091474941318864866534035801

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