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Properties

Label 2-756-7.2-c1-0-4
Degree $2$
Conductor $756$
Sign $0.968 - 0.250i$
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  + (2.5 + 0.866i)7-s + 2·13-s + (0.5 − 0.866i)19-s + (2.5 + 4.33i)25-s + (3.5 + 6.06i)31-s + (5 − 8.66i)37-s + 5·43-s + (5.5 + 4.33i)49-s + (0.5 − 0.866i)61-s + (8 + 13.8i)67-s + (−8.5 − 14.7i)73-s + (2 − 3.46i)79-s + (5 + 1.73i)91-s − 19·97-s + (−10 + 17.3i)103-s + ⋯
L(s)  = 1  + (0.944 + 0.327i)7-s + 0.554·13-s + (0.114 − 0.198i)19-s + (0.5 + 0.866i)25-s + (0.628 + 1.08i)31-s + (0.821 − 1.42i)37-s + 0.762·43-s + (0.785 + 0.618i)49-s + (0.0640 − 0.110i)61-s + (0.977 + 1.69i)67-s + (−0.994 − 1.72i)73-s + (0.225 − 0.389i)79-s + (0.524 + 0.181i)91-s − 1.92·97-s + (−0.985 + 1.70i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73135 + 0.220632i\)
\(L(\frac12)\) \(\approx\) \(1.73135 + 0.220632i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (8.5 + 14.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 19T + 97T^{2} \)
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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

−10.57998483142488946251795659781, −9.373692217032389873932427174036, −8.691483307162343405545450579824, −7.85660714112376026566596534855, −6.96888415483448119798919177665, −5.83843124277115132791998091439, −5.02082628959714852696827090950, −3.98293512923600739671275641037, −2.66876288043943220130448757014, −1.31749937182019740217965767308, 1.14221300684281864038100617277, 2.56330675810612608121570685497, 3.97472129648037134828519000086, 4.79120089437211554270262014208, 5.88427688965774338841376904028, 6.82005589562790945804779948745, 7.926656360406059460528341973197, 8.389802476065613514470183566506, 9.507767734418357731288603926445, 10.37186778999222894972450888345

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