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Properties

Label 2-756-21.5-c1-0-7
Degree $2$
Conductor $756$
Sign $0.553 + 0.832i$
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  + (2 − 1.73i)7-s − 5.19i·13-s + (−3 − 1.73i)19-s + (2.5 + 4.33i)25-s + (7.5 − 4.33i)31-s + (5.5 − 9.52i)37-s − 5·43-s + (1.00 − 6.92i)49-s + (7.5 + 4.33i)61-s + (−2.5 − 4.33i)67-s + (12 − 6.92i)73-s + (−8.5 + 14.7i)79-s + (−9 − 10.3i)91-s + 19.0i·97-s + (−13.5 − 7.79i)103-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)7-s − 1.44i·13-s + (−0.688 − 0.397i)19-s + (0.5 + 0.866i)25-s + (1.34 − 0.777i)31-s + (0.904 − 1.56i)37-s − 0.762·43-s + (0.142 − 0.989i)49-s + (0.960 + 0.554i)61-s + (−0.305 − 0.529i)67-s + (1.40 − 0.810i)73-s + (−0.956 + 1.65i)79-s + (−0.943 − 1.08i)91-s + 1.93i·97-s + (−1.33 − 0.767i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38308 - 0.741119i\)
\(L(\frac12)\) \(\approx\) \(1.38308 - 0.741119i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-7.5 + 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.5 - 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-12 + 6.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−10.38273259269943048516489025902, −9.401251789733498716145497169872, −8.304684175124121524246268673264, −7.75945588857778625781650398968, −6.79978912344790554746277743852, −5.66803436863119794497106666484, −4.78544808537332050285433725828, −3.75286593814605939011564422898, −2.47168693844004023704618418888, −0.869840518111838942871331905076, 1.58818743037637945533912840598, 2.72508431959757112345433782391, 4.25792008117648876543269793641, 4.92986125229453471872497558453, 6.18460482440979899993377330770, 6.85283182151070359654774184826, 8.179722233185757838154967605616, 8.599620532316259353537985511860, 9.610307082363703982920359378759, 10.46207508259370605505636992102

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