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Properties

Label 2-735-105.89-c1-0-16
Degree $2$
Conductor $735$
Sign $-0.991 + 0.132i$
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  + (−0.866 + 1.5i)2-s + (1.72 + 0.158i)3-s + (−0.5 − 0.866i)4-s + (0.358 + 2.20i)5-s + (−1.73 + 2.44i)6-s − 1.73·8-s + (2.94 + 0.548i)9-s + (−3.62 − 1.37i)10-s + (−2.44 + 1.41i)11-s + (−0.724 − 1.57i)12-s − 4·13-s + (0.267 + 3.86i)15-s + (2.49 − 4.33i)16-s + (−2.44 + 1.41i)17-s + (−3.37 + 3.94i)18-s + ⋯
L(s)  = 1  + (−0.612 + 1.06i)2-s + (0.995 + 0.0917i)3-s + (−0.250 − 0.433i)4-s + (0.160 + 0.987i)5-s + (−0.707 + 0.999i)6-s − 0.612·8-s + (0.983 + 0.182i)9-s + (−1.14 − 0.434i)10-s + (−0.738 + 0.426i)11-s + (−0.209 − 0.454i)12-s − 1.10·13-s + (0.0691 + 0.997i)15-s + (0.624 − 1.08i)16-s + (−0.594 + 0.342i)17-s + (−0.795 + 0.930i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.991 + 0.132i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.991 + 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0807496 - 1.21602i\)
\(L(\frac12)\) \(\approx\) \(0.0807496 - 1.21602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.158i)T \)
5 \( 1 + (-0.358 - 2.20i)T \)
7 \( 1 \)
good2 \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (2.44 - 1.41i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (2.44 - 1.41i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.65iT - 29T^{2} \)
31 \( 1 + (-8.48 + 4.89i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 4.89iT - 43T^{2} \)
47 \( 1 + (-2.44 - 1.41i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.46 - 6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.48 - 4.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.24 - 2.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 + (4 + 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−10.29371848960383418273706672232, −9.847782547301176192033651894903, −8.980965770468230504825052139232, −8.026373617571328585707934523222, −7.43858320149163327159211072317, −6.86659973306688153530069204976, −5.81586147809904846468667798350, −4.49394396700352717281484813031, −3.08699013562282674544484558646, −2.30334134601501289000236149856, 0.64218380217712549634649815343, 2.11104269752642505430531209852, 2.76868314733855291396217333384, 4.14165473365001011741292839598, 5.17893914075333163852081385692, 6.52203765707518635120203048478, 7.83719953460635150336304932246, 8.475900557538055687564196054693, 9.181995457967315728945853859457, 9.921361478969588394840530459001

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