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Properties

Label 2-735-105.23-c1-0-48
Degree $2$
Conductor $735$
Sign $-0.994 + 0.106i$
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  + (−2.10 + 0.565i)2-s + (−0.860 − 1.50i)3-s + (2.39 − 1.38i)4-s + (1.79 + 1.32i)5-s + (2.66 + 2.68i)6-s + (−1.18 + 1.18i)8-s + (−1.51 + 2.58i)9-s + (−4.54 − 1.78i)10-s + (−2.93 + 1.69i)11-s + (−4.14 − 2.41i)12-s + (0.206 + 0.206i)13-s + (0.451 − 3.84i)15-s + (−0.935 + 1.62i)16-s + (−0.0612 + 0.228i)17-s + (1.74 − 6.31i)18-s + (−4.60 − 2.65i)19-s + ⋯
L(s)  = 1  + (−1.49 + 0.399i)2-s + (−0.496 − 0.867i)3-s + (1.19 − 0.692i)4-s + (0.804 + 0.594i)5-s + (1.08 + 1.09i)6-s + (−0.419 + 0.419i)8-s + (−0.506 + 0.862i)9-s + (−1.43 − 0.565i)10-s + (−0.883 + 0.510i)11-s + (−1.19 − 0.696i)12-s + (0.0573 + 0.0573i)13-s + (0.116 − 0.993i)15-s + (−0.233 + 0.405i)16-s + (−0.0148 + 0.0554i)17-s + (0.410 − 1.48i)18-s + (−1.05 − 0.609i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.994 + 0.106i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.994 + 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000380380 - 0.00711846i\)
\(L(\frac12)\) \(\approx\) \(0.000380380 - 0.00711846i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.860 + 1.50i)T \)
5 \( 1 + (-1.79 - 1.32i)T \)
7 \( 1 \)
good2 \( 1 + (2.10 - 0.565i)T + (1.73 - i)T^{2} \)
11 \( 1 + (2.93 - 1.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.206 - 0.206i)T + 13iT^{2} \)
17 \( 1 + (0.0612 - 0.228i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.60 + 2.65i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.85 + 6.93i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.84T + 29T^{2} \)
31 \( 1 + (-4.55 - 7.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.92 + 7.19i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.0314iT - 41T^{2} \)
43 \( 1 + (3.76 + 3.76i)T + 43iT^{2} \)
47 \( 1 + (4.87 - 1.30i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.85 + 1.30i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.15 + 8.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.40 - 5.89i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.67 + 2.32i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.95iT - 71T^{2} \)
73 \( 1 + (3.15 - 11.7i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (9.91 + 5.72i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.88 - 3.88i)T - 83iT^{2} \)
89 \( 1 + (-1.00 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.26 - 2.26i)T - 97iT^{2} \)
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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

−10.11456161144795786441573171128, −8.942788920992721318981724052790, −8.260262416729199249796103881757, −7.31713855880288756390351911036, −6.70596050759979394734900438936, −6.04158086346996953233353580677, −4.82046245381307292803753597501, −2.58116537996925218604720634531, −1.70917521630106594523319950337, −0.00619370247678066704434220094, 1.58119192461034978704034357941, 2.95979829495471625800009604006, 4.49185059582433169271957483684, 5.53807491765835447711095091782, 6.31424495137984872631623759875, 7.81894150109917524489727504564, 8.484961391212300093151945739496, 9.318893134958538006735700700152, 9.946989466881171188635825300737, 10.43330879519957674146324071552

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