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Properties

Label 2-756-84.11-c1-0-49
Degree $2$
Conductor $756$
Sign $-0.891 + 0.452i$
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  + (−0.994 − 1.00i)2-s + (−0.0232 + 1.99i)4-s + (1.28 − 0.743i)5-s + (−1.36 − 2.26i)7-s + (2.03 − 1.96i)8-s + (−2.02 − 0.555i)10-s + (1.37 − 2.37i)11-s − 5.10·13-s + (−0.915 + 3.62i)14-s + (−3.99 − 0.0927i)16-s + (4.52 + 2.61i)17-s + (1.30 − 0.754i)19-s + (1.45 + 2.59i)20-s + (−3.75 + 0.983i)22-s + (−4.49 − 7.77i)23-s + ⋯
L(s)  = 1  + (−0.702 − 0.711i)2-s + (−0.0116 + 0.999i)4-s + (0.575 − 0.332i)5-s + (−0.517 − 0.855i)7-s + (0.719 − 0.694i)8-s + (−0.640 − 0.175i)10-s + (0.413 − 0.716i)11-s − 1.41·13-s + (−0.244 + 0.969i)14-s + (−0.999 − 0.0231i)16-s + (1.09 + 0.634i)17-s + (0.299 − 0.172i)19-s + (0.325 + 0.579i)20-s + (−0.800 + 0.209i)22-s + (−0.936 − 1.62i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.891 + 0.452i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.891 + 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183475 - 0.767837i\)
\(L(\frac12)\) \(\approx\) \(0.183475 - 0.767837i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.994 + 1.00i)T \)
3 \( 1 \)
7 \( 1 + (1.36 + 2.26i)T \)
good5 \( 1 + (-1.28 + 0.743i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.37 + 2.37i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.10T + 13T^{2} \)
17 \( 1 + (-4.52 - 2.61i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.30 + 0.754i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.49 + 7.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.11iT - 29T^{2} \)
31 \( 1 + (-0.202 - 0.117i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.50 + 4.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.96iT - 41T^{2} \)
43 \( 1 + 6.52iT - 43T^{2} \)
47 \( 1 + (2.12 + 3.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.44 + 1.98i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.339 + 0.587i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.29 + 9.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.34 - 5.39i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.32T + 71T^{2} \)
73 \( 1 + (1.75 - 3.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (14.4 - 8.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + (-10.8 + 6.27i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−10.09971897935165169112525613322, −9.323697692575162363769748372826, −8.450320815759831389102746882173, −7.51145413355018694984142494500, −6.70781135918470403177119921644, −5.47620655297624646676852758025, −4.17840920298780659581137023167, −3.24484518672228456863384816517, −1.94212490093999196604849138537, −0.49501267818988235390573871802, 1.74496662851074677619954381986, 2.90814282251966851155960896172, 4.72384930467862483992181141902, 5.64800107207146202133443440693, 6.33385622368966945344872468440, 7.35165285869790436511500418682, 7.955146562942798804046659319884, 9.276720618014105672787020727192, 9.843402887293214741233131469037, 10.01383343761719942049666438508

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