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Properties

Label 2-756-36.23-c1-0-7
Degree $2$
Conductor $756$
Sign $-0.532 - 0.846i$
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  + (−1.09 + 0.888i)2-s + (0.419 − 1.95i)4-s + (2.42 + 1.39i)5-s + (−0.866 + 0.5i)7-s + (1.27 + 2.52i)8-s + (−3.90 + 0.614i)10-s + (0.140 + 0.244i)11-s + (−2.43 + 4.22i)13-s + (0.508 − 1.31i)14-s + (−3.64 − 1.64i)16-s + 5.66i·17-s − 7.39i·19-s + (3.74 − 4.14i)20-s + (−0.372 − 0.143i)22-s + (−3.28 + 5.69i)23-s + ⋯
L(s)  = 1  + (−0.777 + 0.628i)2-s + (0.209 − 0.977i)4-s + (1.08 + 0.625i)5-s + (−0.327 + 0.188i)7-s + (0.451 + 0.892i)8-s + (−1.23 + 0.194i)10-s + (0.0425 + 0.0736i)11-s + (−0.676 + 1.17i)13-s + (0.135 − 0.352i)14-s + (−0.911 − 0.410i)16-s + 1.37i·17-s − 1.69i·19-s + (0.838 − 0.927i)20-s + (−0.0793 − 0.0305i)22-s + (−0.685 + 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.532 - 0.846i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.532 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.496670 + 0.898770i\)
\(L(\frac12)\) \(\approx\) \(0.496670 + 0.898770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 - 0.888i)T \)
3 \( 1 \)
7 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (-2.42 - 1.39i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.140 - 0.244i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.43 - 4.22i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.66iT - 17T^{2} \)
19 \( 1 + 7.39iT - 19T^{2} \)
23 \( 1 + (3.28 - 5.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.243 + 0.140i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.09 - 4.67i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.22T + 37T^{2} \)
41 \( 1 + (-0.644 - 0.372i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.31 - 3.64i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.59 + 4.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.11iT - 53T^{2} \)
59 \( 1 + (-0.215 + 0.373i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.95 - 5.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.11 + 1.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 - 0.714T + 73T^{2} \)
79 \( 1 + (-2.15 + 1.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.10 - 7.10i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.354iT - 89T^{2} \)
97 \( 1 + (0.138 + 0.240i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.29765642021843694722798542025, −9.692320806698241170896654447832, −9.097223781826341587367239491557, −8.104169447667905038519999884247, −6.91393856442223297899872041587, −6.50250207113040887009574790510, −5.63792002375471065438274387888, −4.51052749746159184852262240529, −2.68351435976889028049059021411, −1.68345894644729024109504562377, 0.66793523410770579157636533177, 2.09978860401518476202399372261, 3.10665512182573404590048290648, 4.49928536805365624750526382146, 5.62854746296356718145055782400, 6.60466144778821083903990073935, 7.81120815080308329704637702122, 8.391048146048735725443634211679, 9.620280403569251384459817190501, 9.847086226687634719341325096293

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