Properties

Label 2-756-21.5-c1-0-3
Degree $2$
Conductor $756$
Sign $0.744 - 0.667i$
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 + 1.73i)7-s + 1.73i·13-s + (3 + 1.73i)19-s + (2.5 + 4.33i)25-s + (−1.5 + 0.866i)31-s + (−0.5 + 0.866i)37-s + 13·43-s + (1.00 + 6.92i)49-s + (13.5 + 7.79i)61-s + (−5.5 − 9.52i)67-s + (−12 + 6.92i)73-s + (6.5 − 11.2i)79-s + (−2.99 + 3.46i)91-s + 5.19i·97-s + (−16.5 − 9.52i)103-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)7-s + 0.480i·13-s + (0.688 + 0.397i)19-s + (0.5 + 0.866i)25-s + (−0.269 + 0.155i)31-s + (−0.0821 + 0.142i)37-s + 1.98·43-s + (0.142 + 0.989i)49-s + (1.72 + 0.997i)61-s + (−0.671 − 1.16i)67-s + (−1.40 + 0.810i)73-s + (0.731 − 1.26i)79-s + (−0.314 + 0.363i)91-s + 0.527i·97-s + (−1.62 − 0.938i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.744 - 0.667i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53238 + 0.585840i\)
\(L(\frac12)\) \(\approx\) \(1.53238 + 0.585840i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13T + 43T^{2} \)
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 \( 1 + (-13.5 - 7.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (12 - 6.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−10.53221965134939149904171381490, −9.427848159944917642142647213695, −8.825278386768614158005947042004, −7.86676582017030471190609342934, −7.06981337675508768949455224979, −5.87776417435448985938693139540, −5.13269741258244896107698922896, −4.05741181497139106535100787429, −2.74675457327283956911640880157, −1.47871982250095917530485152702, 0.959911025670626443655042082832, 2.50729415977030945124102090545, 3.83043812705157068655464957497, 4.79236386620650775355487539075, 5.71067950860799550435907496887, 6.89937837502719350405088009205, 7.66345269815501757713572860055, 8.444822191514103534905182382007, 9.419036510147655251564257092153, 10.37059750720125497271804065196

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