Properties

Label 2-756-63.41-c1-0-6
Degree $2$
Conductor $756$
Sign $0.0347 + 0.999i$
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  + (2.09 − 3.62i)5-s + (1.36 − 2.26i)7-s + (1.23 − 0.711i)11-s + (−0.850 − 0.491i)13-s − 0.370·17-s + 4.97i·19-s + (−4.98 − 2.87i)23-s + (−6.26 − 10.8i)25-s + (−7.31 + 4.22i)29-s + (6.28 + 3.62i)31-s + (−5.34 − 9.70i)35-s + 3.46·37-s + (1.06 − 1.85i)41-s + (3.00 + 5.21i)43-s + (4.13 + 7.16i)47-s + ⋯
L(s)  = 1  + (0.936 − 1.62i)5-s + (0.517 − 0.855i)7-s + (0.371 − 0.214i)11-s + (−0.235 − 0.136i)13-s − 0.0899·17-s + 1.14i·19-s + (−1.04 − 0.600i)23-s + (−1.25 − 2.17i)25-s + (−1.35 + 0.784i)29-s + (1.12 + 0.651i)31-s + (−0.903 − 1.64i)35-s + 0.569·37-s + (0.167 − 0.289i)41-s + (0.458 + 0.794i)43-s + (0.603 + 1.04i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0347 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0347 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.0347 + 0.999i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.0347 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30772 - 1.26310i\)
\(L(\frac12)\) \(\approx\) \(1.30772 - 1.26310i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 + 2.26i)T \)
good5 \( 1 + (-2.09 + 3.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.23 + 0.711i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.850 + 0.491i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.370T + 17T^{2} \)
19 \( 1 - 4.97iT - 19T^{2} \)
23 \( 1 + (4.98 + 2.87i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.31 - 4.22i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.28 - 3.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.46T + 37T^{2} \)
41 \( 1 + (-1.06 + 1.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.00 - 5.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.13 - 7.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.97iT - 53T^{2} \)
59 \( 1 + (-2.27 + 3.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.50 + 3.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.03 + 8.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + 9.52iT - 73T^{2} \)
79 \( 1 + (-4.25 - 7.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.972 + 1.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.80T + 89T^{2} \)
97 \( 1 + (3.34 - 1.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.971372071800103354276340404445, −9.362871509696854689443163482969, −8.384765104774472471454529141811, −7.83333801249588462885436878781, −6.45315813483251444552178745451, −5.57436964132040099981538041739, −4.72130414224883852960580151082, −3.88602931556353921092951404241, −1.97645322830861893195077828981, −0.976232373705203797822184553158, 2.06221591040560725498430147873, 2.65497968482235000385977176784, 4.04015719403385381842301052030, 5.46963860895414386172634034370, 6.12700803321339536708295229331, 6.99795974838337422119194655153, 7.81760217738367752054123308716, 9.096206746752460171005647791990, 9.711536611721208618139684160330, 10.50962819907928118330208795766

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