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Properties

Label 2-756-28.3-c1-0-43
Degree $2$
Conductor $756$
Sign $0.991 - 0.132i$
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  + (1.38 + 0.296i)2-s + (1.82 + 0.821i)4-s + (0.421 − 0.243i)5-s + (0.250 − 2.63i)7-s + (2.27 + 1.67i)8-s + (0.655 − 0.211i)10-s + (2.51 + 1.45i)11-s + 1.17i·13-s + (1.12 − 3.56i)14-s + (2.65 + 2.99i)16-s + (0.0401 + 0.0231i)17-s + (−2.38 − 4.12i)19-s + (0.969 − 0.0978i)20-s + (3.04 + 2.75i)22-s + (6.02 − 3.47i)23-s + ⋯
L(s)  = 1  + (0.977 + 0.209i)2-s + (0.911 + 0.410i)4-s + (0.188 − 0.108i)5-s + (0.0946 − 0.995i)7-s + (0.805 + 0.592i)8-s + (0.207 − 0.0668i)10-s + (0.757 + 0.437i)11-s + 0.325i·13-s + (0.301 − 0.953i)14-s + (0.662 + 0.748i)16-s + (0.00973 + 0.00561i)17-s + (−0.546 − 0.946i)19-s + (0.216 − 0.0218i)20-s + (0.648 + 0.586i)22-s + (1.25 − 0.725i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.991 - 0.132i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.06266 + 0.203213i\)
\(L(\frac12)\) \(\approx\) \(3.06266 + 0.203213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.296i)T \)
3 \( 1 \)
7 \( 1 + (-0.250 + 2.63i)T \)
good5 \( 1 + (-0.421 + 0.243i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.51 - 1.45i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.17iT - 13T^{2} \)
17 \( 1 + (-0.0401 - 0.0231i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.38 + 4.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.02 + 3.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.465T + 29T^{2} \)
31 \( 1 + (0.536 - 0.928i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.196 - 0.339i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.139iT - 41T^{2} \)
43 \( 1 + 6.47iT - 43T^{2} \)
47 \( 1 + (-3.81 - 6.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.03 - 8.71i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.65 + 6.32i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12.0 - 6.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.49 + 2.01i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.95iT - 71T^{2} \)
73 \( 1 + (7.14 + 4.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (12.0 - 6.92i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.34T + 83T^{2} \)
89 \( 1 + (-9.16 + 5.29i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.51iT - 97T^{2} \)
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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

−10.70594440048922076276272478154, −9.525050426016427295280234197260, −8.591878711067725965281760590633, −7.28955622961549619256541484910, −6.95516475296780372041525216318, −5.91979900638381216117682588511, −4.71085489351007486642317108342, −4.16405616535977045458096742240, −2.95116215522992013299797411038, −1.50578198278645472756044712487, 1.61301288458994532305495805955, 2.80674965076734083786926068949, 3.79351878739173853892994812141, 4.96172981905790031537638786101, 5.87560774226456876807333236197, 6.42535193730279071631173162335, 7.61277647182825095268604108974, 8.648239388299351756454154229419, 9.591901267931719277348964535520, 10.50517791342344899013759237872

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