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Properties

Label 2-1785-5.4-c1-0-54
Degree $2$
Conductor $1785$
Sign $0.388 - 0.921i$
Analytic cond. $14.2532$
Root an. cond. $3.77535$
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  + 1.61i·2-s − i·3-s − 0.608·4-s + (2.06 + 0.867i)5-s + 1.61·6-s − i·7-s + 2.24i·8-s − 9-s + (−1.40 + 3.32i)10-s + 3.70·11-s + 0.608i·12-s − 0.484i·13-s + 1.61·14-s + (0.867 − 2.06i)15-s − 4.84·16-s − i·17-s + ⋯
L(s)  = 1  + 1.14i·2-s − 0.577i·3-s − 0.304·4-s + (0.921 + 0.388i)5-s + 0.659·6-s − 0.377i·7-s + 0.794i·8-s − 0.333·9-s + (−0.443 + 1.05i)10-s + 1.11·11-s + 0.175i·12-s − 0.134i·13-s + 0.431·14-s + (0.224 − 0.532i)15-s − 1.21·16-s − 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1785\)    =    \(3 \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.388 - 0.921i$
Analytic conductor: \(14.2532\)
Root analytic conductor: \(3.77535\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1785} (1429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1785,\ (\ :1/2),\ 0.388 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.471027188\)
\(L(\frac12)\) \(\approx\) \(2.471027188\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (-2.06 - 0.867i)T \)
7 \( 1 + iT \)
17 \( 1 + iT \)
good2 \( 1 - 1.61iT - 2T^{2} \)
11 \( 1 - 3.70T + 11T^{2} \)
13 \( 1 + 0.484iT - 13T^{2} \)
19 \( 1 - 8.59T + 19T^{2} \)
23 \( 1 + 6.32iT - 23T^{2} \)
29 \( 1 - 5.03T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 5.74iT - 37T^{2} \)
41 \( 1 + 4.52T + 41T^{2} \)
43 \( 1 + 0.961iT - 43T^{2} \)
47 \( 1 + 3.97iT - 47T^{2} \)
53 \( 1 + 3.25iT - 53T^{2} \)
59 \( 1 + 0.477T + 59T^{2} \)
61 \( 1 - 1.83T + 61T^{2} \)
67 \( 1 - 3.45iT - 67T^{2} \)
71 \( 1 - 8.74T + 71T^{2} \)
73 \( 1 - 10.6iT - 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 - 5.53iT - 83T^{2} \)
89 \( 1 - 4.43T + 89T^{2} \)
97 \( 1 + 12.0iT - 97T^{2} \)
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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

−9.241567975147860344187497011455, −8.482355936759903620827611752975, −7.58834627300548962176397076121, −6.79283096110699031051814618019, −6.60136425738142760693986500659, −5.58665893637913569100012199845, −4.98183978302497058301683071610, −3.48655472222029071924114872499, −2.38420246397756472316126905504, −1.21866077949614307075561432787, 1.16254302163613460598348687825, 1.97812764802280684224778411588, 3.17360269074546208372165286336, 3.80664602886901444203971388914, 4.97843830843460672473728694035, 5.71370621763195204526570876843, 6.61486243166397282514942285066, 7.60030935868463470088570142019, 8.973510002434140901970739654106, 9.384787482571778517227223591692

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