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Properties

Label 2-756-28.3-c1-0-58
Degree $2$
Conductor $756$
Sign $-0.937 + 0.348i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (0.870 − 1.11i)2-s + (−0.483 − 1.94i)4-s + (−1.71 + 0.988i)5-s + (2.27 + 1.35i)7-s + (−2.58 − 1.15i)8-s + (−0.389 + 2.76i)10-s + (−4.16 − 2.40i)11-s − 6.20i·13-s + (3.48 − 1.35i)14-s + (−3.53 + 1.87i)16-s + (−0.655 − 0.378i)17-s + (−3.65 − 6.33i)19-s + (2.74 + 2.84i)20-s + (−6.30 + 2.54i)22-s + (4.09 − 2.36i)23-s + ⋯
L(s)  = 1  + (0.615 − 0.787i)2-s + (−0.241 − 0.970i)4-s + (−0.765 + 0.441i)5-s + (0.858 + 0.512i)7-s + (−0.913 − 0.407i)8-s + (−0.123 + 0.875i)10-s + (−1.25 − 0.725i)11-s − 1.72i·13-s + (0.932 − 0.361i)14-s + (−0.883 + 0.469i)16-s + (−0.159 − 0.0918i)17-s + (−0.838 − 1.45i)19-s + (0.613 + 0.635i)20-s + (−1.34 + 0.543i)22-s + (0.854 − 0.493i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.937 + 0.348i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.937 + 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.226084 - 1.25520i\)
\(L(\frac12)\) \(\approx\) \(0.226084 - 1.25520i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.870 + 1.11i)T \)
3 \( 1 \)
7 \( 1 + (-2.27 - 1.35i)T \)
good5 \( 1 + (1.71 - 0.988i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.16 + 2.40i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.20iT - 13T^{2} \)
17 \( 1 + (0.655 + 0.378i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.65 + 6.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.09 + 2.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.09T + 29T^{2} \)
31 \( 1 + (-3.85 + 6.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.98 - 6.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.55iT - 41T^{2} \)
43 \( 1 + 1.68iT - 43T^{2} \)
47 \( 1 + (-1.30 - 2.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.08 - 12.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.705 - 1.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.09 - 2.93i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.74 - 2.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + (-4.77 - 2.75i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.07 + 1.77i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.69T + 83T^{2} \)
89 \( 1 + (-3.63 + 2.09i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.34iT - 97T^{2} \)
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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

−10.48550403934473927385541450802, −9.150375851549884440148695676530, −8.239460917887425880503192766640, −7.52782092953700438537570497000, −6.10966045778806558337178900165, −5.25661136868360347413224495627, −4.49508695389808707244809528956, −3.09479823041289240581649969383, −2.53579374249763224477692874664, −0.51492739997048549537843378179, 2.01716332333906886603812027827, 3.74533093570806019292292109035, 4.51561720818196407978283439154, 5.09433681528324801138188819240, 6.42233751579499962099804375782, 7.34459851576459385813423196071, 7.983298512661762012935303796854, 8.611844094365053201326264230535, 9.749896369332129905313830005283, 10.95565402775104442082868545500

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