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Properties

Label 2-756-21.5-c1-0-8
Degree $2$
Conductor $756$
Sign $-0.0633 + 0.997i$
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  + (1.07 − 1.87i)5-s + (−0.167 − 2.64i)7-s + (1.15 − 0.667i)11-s + 2.35i·13-s + (−3.60 − 6.23i)17-s + (1.03 + 0.599i)19-s + (−2.08 − 1.20i)23-s + (0.167 + 0.290i)25-s − 6.14i·29-s + (−7.11 + 4.10i)31-s + (−5.11 − 2.53i)35-s + (2.57 − 4.45i)37-s + 4.47·41-s − 0.664·43-s + (1.07 − 1.87i)47-s + ⋯
L(s)  = 1  + (0.482 − 0.836i)5-s + (−0.0633 − 0.997i)7-s + (0.348 − 0.201i)11-s + 0.651i·13-s + (−0.873 − 1.51i)17-s + (0.238 + 0.137i)19-s + (−0.434 − 0.250i)23-s + (0.0335 + 0.0580i)25-s − 1.14i·29-s + (−1.27 + 0.737i)31-s + (−0.865 − 0.429i)35-s + (0.423 − 0.732i)37-s + 0.698·41-s − 0.101·43-s + (0.157 − 0.272i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.0633 + 0.997i$
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: \((2,\ 756,\ (\ :1/2),\ -0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00760 - 1.07357i\)
\(L(\frac12)\) \(\approx\) \(1.00760 - 1.07357i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.167 + 2.64i)T \)
good5 \( 1 + (-1.07 + 1.87i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.15 + 0.667i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.35iT - 13T^{2} \)
17 \( 1 + (3.60 + 6.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.03 - 0.599i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.08 + 1.20i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.14iT - 29T^{2} \)
31 \( 1 + (7.11 - 4.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.57 + 4.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 0.664T + 43T^{2} \)
47 \( 1 + (-1.07 + 1.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.0 + 6.37i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.83 + 8.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.6 - 6.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.86 - 3.23i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (-9.07 + 5.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.53 + 7.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + (2.59 - 4.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.23iT - 97T^{2} \)
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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

−9.921856432516182643422935403824, −9.324885465221686653416988379607, −8.585176633642811742978199749862, −7.39734672896427689250302587563, −6.74061915100243573359266640589, −5.56841227377771347330458862527, −4.63355171650739166062130872390, −3.77719134857960334545075956879, −2.17384532701422359632799595367, −0.75448469511616770308968461905, 1.88371535646793331304621111520, 2.86760795743681764799666894702, 4.04521916702958879924467293613, 5.45248910682288948708726811368, 6.13201688797941501196063365988, 6.93529002262294263669703094292, 8.070166837853897848318852702621, 8.913476490479850549904994913515, 9.718859219474402557299540044801, 10.63084960077960522955833374739

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