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Properties

Label 2-735-105.32-c1-0-61
Degree $2$
Conductor $735$
Sign $0.996 + 0.0821i$
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  + (2.45 + 0.658i)2-s + (1.25 − 1.19i)3-s + (3.87 + 2.23i)4-s + (−1.98 − 1.02i)5-s + (3.87 − 2.10i)6-s + (4.45 + 4.45i)8-s + (0.154 − 2.99i)9-s + (−4.21 − 3.82i)10-s + (1.35 + 0.784i)11-s + (7.53 − 1.81i)12-s + (2.21 − 2.21i)13-s + (−3.71 + 1.08i)15-s + (3.54 + 6.14i)16-s + (1.32 + 4.92i)17-s + (2.35 − 7.26i)18-s + (−1.45 + 0.840i)19-s + ⋯
L(s)  = 1  + (1.73 + 0.465i)2-s + (0.725 − 0.688i)3-s + (1.93 + 1.11i)4-s + (−0.889 − 0.457i)5-s + (1.58 − 0.859i)6-s + (1.57 + 1.57i)8-s + (0.0514 − 0.998i)9-s + (−1.33 − 1.20i)10-s + (0.409 + 0.236i)11-s + (2.17 − 0.523i)12-s + (0.615 − 0.615i)13-s + (−0.959 + 0.280i)15-s + (0.886 + 1.53i)16-s + (0.320 + 1.19i)17-s + (0.554 − 1.71i)18-s + (−0.333 + 0.192i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.996 + 0.0821i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.996 + 0.0821i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.58546 - 0.188732i\)
\(L(\frac12)\) \(\approx\) \(4.58546 - 0.188732i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.25 + 1.19i)T \)
5 \( 1 + (1.98 + 1.02i)T \)
7 \( 1 \)
good2 \( 1 + (-2.45 - 0.658i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-1.35 - 0.784i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.21 + 2.21i)T - 13iT^{2} \)
17 \( 1 + (-1.32 - 4.92i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.45 - 0.840i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.364 + 1.36i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 8.91T + 29T^{2} \)
31 \( 1 + (1.37 - 2.38i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.161 + 0.601i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.44iT - 41T^{2} \)
43 \( 1 + (5.47 - 5.47i)T - 43iT^{2} \)
47 \( 1 + (5.04 + 1.35i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.87 - 1.03i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.77 - 4.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.70 + 6.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.12 + 1.37i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.61iT - 71T^{2} \)
73 \( 1 + (2.15 + 8.05i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-14.7 + 8.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.21 - 3.21i)T + 83iT^{2} \)
89 \( 1 + (-4.70 - 8.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.39 - 4.39i)T + 97iT^{2} \)
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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

−10.79958357479063302741284375247, −9.228951597639528840019047153226, −8.110789597293399579109261725761, −7.74214610502363585857344106210, −6.66189668944280246182911558317, −5.98484432841818124818915965493, −4.79825380133964109630725755994, −3.75285287907373876619195950761, −3.30788807942689594575852932747, −1.71234318339396330752649275581, 2.08650524078025522169361239509, 3.27287460677019732954224491264, 3.77041814994559503210457929969, 4.60342874043418201147752433483, 5.55246885252355187553896147293, 6.75419005591387220136893255718, 7.54015341062642845736467947848, 8.755422347196559509658886645476, 9.741810216200473448942833050379, 10.87494847407901322895961424838

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