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Properties

Label 2-756-7.4-c1-0-1
Degree $2$
Conductor $756$
Sign $0.386 - 0.922i$
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  + (−1.58 − 2.73i)5-s + (0.5 + 2.59i)7-s + (−3.16 + 5.47i)11-s + (1.58 − 2.73i)17-s + (3.5 + 6.06i)19-s + (1.58 + 2.73i)23-s + (−2.5 + 4.33i)25-s + 3.16·29-s + (−1.5 + 2.59i)31-s + (6.32 − 5.47i)35-s + (2 + 3.46i)37-s + 9.48·41-s + 5·43-s + (−4.74 − 8.21i)47-s + (−6.5 + 2.59i)49-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)5-s + (0.188 + 0.981i)7-s + (−0.953 + 1.65i)11-s + (0.383 − 0.664i)17-s + (0.802 + 1.39i)19-s + (0.329 + 0.571i)23-s + (−0.5 + 0.866i)25-s + 0.587·29-s + (−0.269 + 0.466i)31-s + (1.06 − 0.925i)35-s + (0.328 + 0.569i)37-s + 1.48·41-s + 0.762·43-s + (−0.691 − 1.19i)47-s + (−0.928 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904508 + 0.601662i\)
\(L(\frac12)\) \(\approx\) \(0.904508 + 0.601662i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (1.58 + 2.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.16 - 5.47i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-1.58 + 2.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.58 - 2.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.16T + 29T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.48T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (4.74 + 8.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.74 - 8.21i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.32 - 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.32T + 83T^{2} \)
89 \( 1 + (-4.74 - 8.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−10.34749419796380299295395895581, −9.538286160425849559605053745072, −8.827053768182597813791237592951, −7.82189208755716915296583368332, −7.43670066713073198658901004969, −5.78719686531878836303649301820, −5.04660747963882135909051330952, −4.35121478697173678711743405307, −2.85667905731729881285241079692, −1.46666538950011298566897024351, 0.58959884735531724501078841954, 2.81923699543565944240662170460, 3.44250531904788190475577079798, 4.60682683621401214865326235609, 5.88164196817968200826309463182, 6.79424738633053806223789997352, 7.66834128324475105431123091609, 8.149620087364573471917294241758, 9.406636879509823684727153522459, 10.56001863260197186318171396863

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