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Properties

Label 2-756-7.6-c0-0-1
Degree $2$
Conductor $756$
Sign $0.866 - 0.5i$
Analytic cond. $0.377293$
Root an. cond. $0.614241$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)7-s + 1.73i·13-s − 1.73i·19-s + 25-s + 37-s − 2·43-s + (−0.499 + 0.866i)49-s − 1.73i·61-s − 67-s − 1.73i·73-s − 79-s + (−1.49 + 0.866i)91-s + 1.73i·97-s − 1.73i·103-s − 2·109-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + 1.73i·13-s − 1.73i·19-s + 25-s + 37-s − 2·43-s + (−0.499 + 0.866i)49-s − 1.73i·61-s − 67-s − 1.73i·73-s − 79-s + (−1.49 + 0.866i)91-s + 1.73i·97-s − 1.73i·103-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(0.377293\)
Root analytic conductor: \(0.614241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :0),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.006702706\)
\(L(\frac12)\) \(\approx\) \(1.006702706\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.73iT - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−10.81627331308106268841368742117, −9.481118495532239492645039493986, −9.032729041608048254449298649532, −8.203448041748762946636591955442, −7.00584171065874076699269276637, −6.37124664909561754818811363865, −5.07614452035563599567262492905, −4.45355542891509944994073311167, −2.92899972593618670672730398565, −1.81187151150129448345155737362, 1.28792710645896344655266296786, 2.98787246350130574352395025997, 4.00547678673688042860799485869, 5.12175397969933543723655827394, 5.98297523755242811989061698964, 7.15803168256025251231831132106, 7.958439179176515890108735119133, 8.522773572496427259998880095602, 9.992166741355502453753406247897, 10.32330226223228146695408613789

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