Properties

Label 2-756-7.4-c1-0-10
Degree $2$
Conductor $756$
Sign $-0.911 + 0.411i$
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  + (−2.14 − 3.71i)5-s + (1.40 − 2.24i)7-s + (−1.90 + 3.30i)11-s + 3.28·13-s + (−0.405 + 0.702i)17-s + (−3.54 − 6.14i)19-s + (−3.23 − 5.60i)23-s + (−6.69 + 11.5i)25-s − 3.81·29-s + (−1.64 + 2.84i)31-s + (−11.3 − 0.413i)35-s + (2.88 + 4.99i)37-s − 2.09·41-s − 8.76·43-s + (1.66 + 2.88i)47-s + ⋯
L(s)  = 1  + (−0.958 − 1.66i)5-s + (0.531 − 0.847i)7-s + (−0.574 + 0.995i)11-s + 0.911·13-s + (−0.0983 + 0.170i)17-s + (−0.814 − 1.41i)19-s + (−0.675 − 1.16i)23-s + (−1.33 + 2.31i)25-s − 0.707·29-s + (−0.295 + 0.511i)31-s + (−1.91 − 0.0698i)35-s + (0.473 + 0.820i)37-s − 0.327·41-s − 1.33·43-s + (0.243 + 0.421i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.911 + 0.411i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.911 + 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.176622 - 0.821015i\)
\(L(\frac12)\) \(\approx\) \(0.176622 - 0.821015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 2.24i)T \)
good5 \( 1 + (2.14 + 3.71i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.90 - 3.30i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.28T + 13T^{2} \)
17 \( 1 + (0.405 - 0.702i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.54 + 6.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.23 + 5.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.81T + 29T^{2} \)
31 \( 1 + (1.64 - 2.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.88 - 4.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.09T + 41T^{2} \)
43 \( 1 + 8.76T + 43T^{2} \)
47 \( 1 + (-1.66 - 2.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.93 + 8.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.73 + 3.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.97 + 5.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.76 + 3.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.05T + 71T^{2} \)
73 \( 1 + (-5.19 + 8.99i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.57 - 4.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.71T + 83T^{2} \)
89 \( 1 + (-6.26 - 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.04T + 97T^{2} \)
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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

−9.970768820257272654255195734983, −8.852276224660831467290255083062, −8.317310902327643618157249926301, −7.60268739846125142553707236974, −6.58832155949694652548789441255, −5.01862145242924808445899959582, −4.60743350287344586503040100912, −3.75535235404111042530251109991, −1.79236073353234755384242433461, −0.42184164588472040191055514610, 2.14043023330103465195535916081, 3.31180128413584180940656848660, 3.97622422468823925535762190792, 5.70353570005274622091750501652, 6.17459397727687555999343201396, 7.44510822250563360814676744662, 8.025088639875266949740466667964, 8.781323927432560547920974152563, 10.15404267434762176090316176522, 10.81970813279401433812018032087

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