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Properties

Label 2-735-105.2-c1-0-0
Degree $2$
Conductor $735$
Sign $-0.784 + 0.620i$
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  + (0.0952 + 0.355i)2-s + (−0.904 + 1.47i)3-s + (1.61 − 0.932i)4-s + (−2.22 − 0.249i)5-s + (−0.611 − 0.180i)6-s + (1.00 + 1.00i)8-s + (−1.36 − 2.67i)9-s + (−0.123 − 0.813i)10-s + (−2.93 + 1.69i)11-s + (−0.0835 + 3.22i)12-s + (−1.59 + 1.59i)13-s + (2.37 − 3.05i)15-s + (1.60 − 2.77i)16-s + (0.192 + 0.0514i)17-s + (0.820 − 0.739i)18-s + (−6.36 − 3.67i)19-s + ⋯
L(s)  = 1  + (0.0673 + 0.251i)2-s + (−0.522 + 0.852i)3-s + (0.807 − 0.466i)4-s + (−0.993 − 0.111i)5-s + (−0.249 − 0.0738i)6-s + (0.355 + 0.355i)8-s + (−0.454 − 0.890i)9-s + (−0.0389 − 0.257i)10-s + (−0.884 + 0.510i)11-s + (−0.0241 + 0.931i)12-s + (−0.442 + 0.442i)13-s + (0.614 − 0.789i)15-s + (0.400 − 0.694i)16-s + (0.0466 + 0.0124i)17-s + (0.193 − 0.174i)18-s + (−1.45 − 0.842i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.784 + 0.620i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.784 + 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00984992 - 0.0283172i\)
\(L(\frac12)\) \(\approx\) \(0.00984992 - 0.0283172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.904 - 1.47i)T \)
5 \( 1 + (2.22 + 0.249i)T \)
7 \( 1 \)
good2 \( 1 + (-0.0952 - 0.355i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (2.93 - 1.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.59 - 1.59i)T - 13iT^{2} \)
17 \( 1 + (-0.192 - 0.0514i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.36 + 3.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.02 - 0.810i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.49T + 29T^{2} \)
31 \( 1 + (-0.461 - 0.798i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.08 - 2.16i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 1.39iT - 41T^{2} \)
43 \( 1 + (-0.864 + 0.864i)T - 43iT^{2} \)
47 \( 1 + (0.238 + 0.889i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.39 + 8.93i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.12 + 5.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.916 + 1.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.298 - 1.11i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.77iT - 71T^{2} \)
73 \( 1 + (6.56 + 1.75i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.95 - 1.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.26 - 6.26i)T + 83iT^{2} \)
89 \( 1 + (-6.18 + 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.71 - 6.71i)T + 97iT^{2} \)
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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

−10.83956309763427997924623585732, −10.31217685660290152270392891006, −9.305012236196579790376287776155, −8.261435664303229646101717109640, −7.26837842683687615576567038305, −6.56455464036065388120264975881, −5.39027155038289718571806220552, −4.71680176755245323699859966670, −3.66314217450496483141716276974, −2.23467387135045866778176249502, 0.01443021631205261738864403452, 1.93680612127919005319605471594, 3.02099991716254585710457126332, 4.15625218826264314239217797273, 5.55053374579357617728856339882, 6.41786561921948936179154296683, 7.46739591704431116020144860156, 7.80812011332046286983289986637, 8.624110641441401389582433294023, 10.47318135216093984946391302675

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