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Properties

Label 2-735-105.23-c1-0-2
Degree $2$
Conductor $735$
Sign $-0.783 + 0.621i$
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  + (0.474 − 0.127i)2-s + (0.708 + 1.58i)3-s + (−1.52 + 0.879i)4-s + (−2.23 + 0.0580i)5-s + (0.536 + 0.659i)6-s + (−1.30 + 1.30i)8-s + (−1.99 + 2.24i)9-s + (−1.05 + 0.311i)10-s + (2.31 − 1.33i)11-s + (−2.46 − 1.78i)12-s + (−2.14 − 2.14i)13-s + (−1.67 − 3.49i)15-s + (1.30 − 2.26i)16-s + (−1.19 + 4.46i)17-s + (−0.661 + 1.31i)18-s + (−4.54 − 2.62i)19-s + ⋯
L(s)  = 1  + (0.335 − 0.0898i)2-s + (0.409 + 0.912i)3-s + (−0.761 + 0.439i)4-s + (−0.999 + 0.0259i)5-s + (0.219 + 0.269i)6-s + (−0.461 + 0.461i)8-s + (−0.665 + 0.746i)9-s + (−0.332 + 0.0984i)10-s + (0.697 − 0.402i)11-s + (−0.712 − 0.515i)12-s + (−0.596 − 0.596i)13-s + (−0.432 − 0.901i)15-s + (0.326 − 0.565i)16-s + (−0.290 + 1.08i)17-s + (−0.155 + 0.310i)18-s + (−1.04 − 0.601i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.783 + 0.621i$
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: \((2,\ 735,\ (\ :1/2),\ -0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0711015 - 0.203897i\)
\(L(\frac12)\) \(\approx\) \(0.0711015 - 0.203897i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.708 - 1.58i)T \)
5 \( 1 + (2.23 - 0.0580i)T \)
7 \( 1 \)
good2 \( 1 + (-0.474 + 0.127i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-2.31 + 1.33i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.14 + 2.14i)T + 13iT^{2} \)
17 \( 1 + (1.19 - 4.46i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.54 + 2.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.932 + 3.48i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.86T + 29T^{2} \)
31 \( 1 + (2.64 + 4.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.784 + 2.92i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 11.5iT - 41T^{2} \)
43 \( 1 + (-0.759 - 0.759i)T + 43iT^{2} \)
47 \( 1 + (10.4 - 2.80i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.05 + 1.62i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.0797 - 0.138i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.36 + 4.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.39 - 1.98i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (1.52 - 5.68i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.37 - 1.94i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.03 - 4.03i)T - 83iT^{2} \)
89 \( 1 + (-1.97 + 3.42i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.86 + 1.86i)T - 97iT^{2} \)
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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

−11.03669513467290968262692554703, −9.962860845484752508266534849496, −9.087014956356051299041552762884, −8.333967778794565978880684855549, −7.890579895201887210209427246077, −6.41739986591686545624801691988, −5.13533154709833785086291683403, −4.26515325848843354577050457594, −3.75386910330076629268291497410, −2.70341084412051124621407716994, 0.094761787576987821527042633521, 1.75126930186022261815463491893, 3.38832420351982770024336276161, 4.24676998679283941579963824861, 5.25689054573318403276658250781, 6.54077218902372834585875457526, 7.14987312062344512124872732923, 8.121014361633173693781721737431, 8.996297047159592005596791617210, 9.536548783522818726879480994732

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