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Properties

Label 2-756-12.11-c1-0-15
Degree $2$
Conductor $756$
Sign $0.903 + 0.428i$
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  + (0.310 − 1.37i)2-s + (−1.80 − 0.856i)4-s + 1.05i·5-s + i·7-s + (−1.74 + 2.22i)8-s + (1.45 + 0.327i)10-s + 2.97·11-s + 0.985·13-s + (1.37 + 0.310i)14-s + (2.53 + 3.09i)16-s + 6.35i·17-s − 1.26i·19-s + (0.902 − 1.90i)20-s + (0.924 − 4.11i)22-s + 2.66·23-s + ⋯
L(s)  = 1  + (0.219 − 0.975i)2-s + (−0.903 − 0.428i)4-s + 0.471i·5-s + 0.377i·7-s + (−0.616 + 0.787i)8-s + (0.459 + 0.103i)10-s + 0.898·11-s + 0.273·13-s + (0.368 + 0.0829i)14-s + (0.633 + 0.773i)16-s + 1.54i·17-s − 0.291i·19-s + (0.201 − 0.425i)20-s + (0.197 − 0.876i)22-s + 0.555·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.903 + 0.428i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.903 + 0.428i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54495 - 0.347564i\)
\(L(\frac12)\) \(\approx\) \(1.54495 - 0.347564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.310 + 1.37i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 1.05iT - 5T^{2} \)
11 \( 1 - 2.97T + 11T^{2} \)
13 \( 1 - 0.985T + 13T^{2} \)
17 \( 1 - 6.35iT - 17T^{2} \)
19 \( 1 + 1.26iT - 19T^{2} \)
23 \( 1 - 2.66T + 23T^{2} \)
29 \( 1 - 5.12iT - 29T^{2} \)
31 \( 1 + 8.41iT - 31T^{2} \)
37 \( 1 - 8.91T + 37T^{2} \)
41 \( 1 - 4.54iT - 41T^{2} \)
43 \( 1 + 0.646iT - 43T^{2} \)
47 \( 1 - 4.64T + 47T^{2} \)
53 \( 1 - 2.57iT - 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 - 4.89T + 61T^{2} \)
67 \( 1 + 8.21iT - 67T^{2} \)
71 \( 1 - 0.583T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 15.1iT - 79T^{2} \)
83 \( 1 - 4.73T + 83T^{2} \)
89 \( 1 + 1.74iT - 89T^{2} \)
97 \( 1 - 3.00T + 97T^{2} \)
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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

−10.53205479021305023665311935454, −9.465070466709397160954717992109, −8.880570053590508859842661517865, −7.912584767137477857923924407277, −6.52711645765977460279544503636, −5.82714920706973048006955464820, −4.56258316262042136120750777048, −3.67166273944972430639666410286, −2.62811142033176985671258122229, −1.33583877459606582819409325666, 0.920171833893575728589927260319, 3.10229722654224246656870436305, 4.27658657304293479992358042570, 4.99225230142961097454724260017, 6.06249410748164402810431263101, 6.94743042826242250415747006538, 7.63634803310513592379963462992, 8.760856907130298768581941160615, 9.209270713256741736039241581712, 10.16143303713910622855506876233

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