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Properties

Label 2-756-21.17-c1-0-9
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: $\chi_{756} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.830269 - 0.779251i\)
\(L(\frac12)\) \(\approx\) \(0.830269 - 0.779251i\)
\(L(\frac{3}{2})\) 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

−9.871432544365117460050648076408, −9.348962664314911990725600715408, −8.294454273461143192755299695428, −7.85774013787530520712301106297, −6.60613329542941072301751035371, −5.35793773748294744898380731072, −5.00778499296106911112240543793, −3.52621170726714653268024508603, −2.44246441702264429816000747619, −0.60587536992446970911642452948, 1.55550706521270161200005757834, 3.26653326307454291170080140409, 3.87888865415167023421973513765, 5.12378218332558943926205051581, 6.33633728189808823684220633735, 7.21919692756864165205823721697, 7.69788235210429034496438296686, 8.824014908352098406980497660458, 9.881888396950446175227228391348, 10.74271560850803757665613054024

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