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Properties

Label 2-756-28.27-c1-0-61
Degree $2$
Conductor $756$
Sign $-0.987 + 0.157i$
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  + (0.840 − 1.13i)2-s + (−0.588 − 1.91i)4-s − 2.67i·5-s + (−0.370 + 2.61i)7-s + (−2.66 − 0.935i)8-s + (−3.04 − 2.25i)10-s − 1.39i·11-s − 3.08i·13-s + (2.66 + 2.62i)14-s + (−3.30 + 2.25i)16-s − 5.83i·17-s − 1.91·19-s + (−5.12 + 1.57i)20-s + (−1.58 − 1.17i)22-s + 1.39i·23-s + ⋯
L(s)  = 1  + (0.593 − 0.804i)2-s + (−0.294 − 0.955i)4-s − 1.19i·5-s + (−0.140 + 0.990i)7-s + (−0.943 − 0.330i)8-s + (−0.963 − 0.711i)10-s − 0.421i·11-s − 0.855i·13-s + (0.713 + 0.700i)14-s + (−0.826 + 0.562i)16-s − 1.41i·17-s − 0.440·19-s + (−1.14 + 0.352i)20-s + (−0.338 − 0.250i)22-s + 0.291i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.987 + 0.157i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.987 + 0.157i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.122635 - 1.54656i\)
\(L(\frac12)\) \(\approx\) \(0.122635 - 1.54656i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.840 + 1.13i)T \)
3 \( 1 \)
7 \( 1 + (0.370 - 2.61i)T \)
good5 \( 1 + 2.67iT - 5T^{2} \)
11 \( 1 + 1.39iT - 11T^{2} \)
13 \( 1 + 3.08iT - 13T^{2} \)
17 \( 1 + 5.83iT - 17T^{2} \)
19 \( 1 + 1.91T + 19T^{2} \)
23 \( 1 - 1.39iT - 23T^{2} \)
29 \( 1 + 4.90T + 29T^{2} \)
31 \( 1 + 9.53T + 31T^{2} \)
37 \( 1 - 6.83T + 37T^{2} \)
41 \( 1 + 0.693iT - 41T^{2} \)
43 \( 1 - 0.678iT - 43T^{2} \)
47 \( 1 - 8.26T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 - 12.8iT - 61T^{2} \)
67 \( 1 + 3.08iT - 67T^{2} \)
71 \( 1 + 9.79iT - 71T^{2} \)
73 \( 1 + 5.91iT - 73T^{2} \)
79 \( 1 + 12.0iT - 79T^{2} \)
83 \( 1 + 5.77T + 83T^{2} \)
89 \( 1 - 6.65iT - 89T^{2} \)
97 \( 1 + 12.8iT - 97T^{2} \)
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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

−9.899315253560473055302655472373, −9.080277763690591492048958151587, −8.692840457910745797915411833986, −7.34941440394538462765464423246, −5.76380487775306948958824863191, −5.47278376140292331831087165482, −4.47822548910558256735080728535, −3.26163825773620036708597661259, −2.15718524570644720074794704573, −0.63699145618300570659484359183, 2.27169606958766619220203513245, 3.73570668547176715776893007636, 4.13131161275769672827042220366, 5.59189057427683320722801024358, 6.59169487098454443883793303962, 7.04729840536581679639101147968, 7.81585189976754459227601488173, 8.895036368032323994941182320279, 9.955395199557700262069264958210, 10.82909440109177000707576024795

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