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Properties

Label 2-756-28.19-c1-0-50
Degree $2$
Conductor $756$
Sign $0.716 + 0.697i$
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  + (1.15 − 0.820i)2-s + (0.655 − 1.88i)4-s + (2.47 + 1.42i)5-s + (2.58 − 0.555i)7-s + (−0.794 − 2.71i)8-s + (4.01 − 0.382i)10-s + (1.53 − 0.884i)11-s + 4.22i·13-s + (2.52 − 2.76i)14-s + (−3.14 − 2.47i)16-s + (−5.64 + 3.25i)17-s + (0.271 − 0.470i)19-s + (4.31 − 3.73i)20-s + (1.04 − 2.27i)22-s + (2.52 + 1.45i)23-s + ⋯
L(s)  = 1  + (0.814 − 0.579i)2-s + (0.327 − 0.944i)4-s + (1.10 + 0.637i)5-s + (0.977 − 0.210i)7-s + (−0.281 − 0.959i)8-s + (1.27 − 0.120i)10-s + (0.462 − 0.266i)11-s + 1.17i·13-s + (0.674 − 0.738i)14-s + (−0.785 − 0.618i)16-s + (−1.36 + 0.790i)17-s + (0.0623 − 0.107i)19-s + (0.964 − 0.835i)20-s + (0.221 − 0.485i)22-s + (0.525 + 0.303i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86521 - 1.16504i\)
\(L(\frac12)\) \(\approx\) \(2.86521 - 1.16504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 + 0.820i)T \)
3 \( 1 \)
7 \( 1 + (-2.58 + 0.555i)T \)
good5 \( 1 + (-2.47 - 1.42i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.53 + 0.884i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.22iT - 13T^{2} \)
17 \( 1 + (5.64 - 3.25i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.271 + 0.470i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.52 - 1.45i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.89T + 29T^{2} \)
31 \( 1 + (3.59 + 6.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.62 + 6.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + 8.47iT - 43T^{2} \)
47 \( 1 + (0.518 - 0.897i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.96 + 5.14i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.373 - 0.646i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.14 + 1.81i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.4 - 7.78i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.60iT - 71T^{2} \)
73 \( 1 + (-7.52 + 4.34i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.876 - 0.506i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + (-6.49 - 3.74i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.23iT - 97T^{2} \)
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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

−10.50147363723614681478987627096, −9.512287927390035336401030577466, −8.874197611793910199401060066371, −7.33444896730450314959490649734, −6.45115206476432146659174904755, −5.80264274785470755628101068053, −4.65588176778982842815447563973, −3.83745150088761455141434926491, −2.31295447164421822555208193327, −1.67076531789894157140191283119, 1.70258476442651830635019780377, 2.87145474061339567131179054611, 4.43070229988572156826822796526, 5.12374984881988032392126670214, 5.79125116787559748598973141298, 6.82098619641296189779727377819, 7.76592310170244052936113394865, 8.741909810849843942342231823604, 9.284263759052925513055956935801, 10.61443643643821545349128171073

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