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Properties

Label 2-756-84.23-c1-0-54
Degree $2$
Conductor $756$
Sign $-0.900 + 0.434i$
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  + (−1.33 + 0.476i)2-s + (1.54 − 1.26i)4-s + (2.47 + 1.43i)5-s + (−2.52 − 0.784i)7-s + (−1.45 + 2.42i)8-s + (−3.97 − 0.724i)10-s + (−0.887 − 1.53i)11-s − 4.70·13-s + (3.73 − 0.159i)14-s + (0.781 − 3.92i)16-s + (−6.08 + 3.51i)17-s + (−6.93 − 4.00i)19-s + (5.64 − 0.930i)20-s + (1.91 + 1.62i)22-s + (2.46 − 4.26i)23-s + ⋯
L(s)  = 1  + (−0.941 + 0.336i)2-s + (0.773 − 0.634i)4-s + (1.10 + 0.639i)5-s + (−0.955 − 0.296i)7-s + (−0.514 + 0.857i)8-s + (−1.25 − 0.229i)10-s + (−0.267 − 0.463i)11-s − 1.30·13-s + (0.999 − 0.0426i)14-s + (0.195 − 0.980i)16-s + (−1.47 + 0.852i)17-s + (−1.59 − 0.918i)19-s + (1.26 − 0.208i)20-s + (0.407 + 0.346i)22-s + (0.513 − 0.889i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.900 + 0.434i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.900 + 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00138279 - 0.00604820i\)
\(L(\frac12)\) \(\approx\) \(0.00138279 - 0.00604820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.476i)T \)
3 \( 1 \)
7 \( 1 + (2.52 + 0.784i)T \)
good5 \( 1 + (-2.47 - 1.43i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.887 + 1.53i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.70T + 13T^{2} \)
17 \( 1 + (6.08 - 3.51i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.93 + 4.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.46 + 4.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.21iT - 29T^{2} \)
31 \( 1 + (1.31 - 0.757i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.38 + 2.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.94iT - 41T^{2} \)
43 \( 1 - 7.39iT - 43T^{2} \)
47 \( 1 + (2.78 - 4.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.87 - 1.66i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.717 - 1.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.30 + 12.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.58 + 1.49i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + (-0.0244 - 0.0423i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.15 + 0.668i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.02T + 83T^{2} \)
89 \( 1 + (5.61 + 3.24i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.07T + 97T^{2} \)
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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

−9.939421910357919183885176166145, −9.167127804357661385400068500194, −8.502115232151133182980939398102, −7.10027371049980034710994225713, −6.62170388207885427361977898780, −5.98910305557622596622325492432, −4.68906138726933022364846086970, −2.84039963652591438407731081945, −2.12387586763060230908486408548, −0.00371969660578287955272506519, 1.97634975241014948769406732046, 2.62745310835001644879769080388, 4.27269219200961688787314196631, 5.54027436428712664974939517995, 6.50422833636116936138738298985, 7.25012725134695135197389099475, 8.457534725910906052998816881791, 9.223539507038693413394525421178, 9.801534082940865060765298107029, 10.26194505496670673306888938512

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