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Properties

Label 2-28e2-7.4-c1-0-9
Degree $2$
Conductor $784$
Sign $0.991 - 0.126i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
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  + (1.5 + 2.59i)9-s + (2 − 3.46i)11-s + (4 + 6.92i)23-s + (2.5 − 4.33i)25-s + 2·29-s + (3 + 5.19i)37-s + 12·43-s + (5 − 8.66i)53-s + (2 − 3.46i)67-s − 16·71-s + (4 + 6.92i)79-s + (−4.5 + 7.79i)81-s + 12·99-s + (−10 − 17.3i)107-s + (−9 + 15.5i)109-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)9-s + (0.603 − 1.04i)11-s + (0.834 + 1.44i)23-s + (0.5 − 0.866i)25-s + 0.371·29-s + (0.493 + 0.854i)37-s + 1.82·43-s + (0.686 − 1.18i)53-s + (0.244 − 0.423i)67-s − 1.89·71-s + (0.450 + 0.779i)79-s + (−0.5 + 0.866i)81-s + 1.20·99-s + (−0.966 − 1.67i)107-s + (−0.862 + 1.49i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67093 + 0.106037i\)
\(L(\frac12)\) \(\approx\) \(1.67093 + 0.106037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.38187217000862441395582993407, −9.443538981855195754142310424490, −8.601496357107932162719582229730, −7.76976371199354797625952189785, −6.86917580118826050726950422152, −5.88597164984551868472163443226, −4.93281723194342072687458241464, −3.87753582944241538986659644053, −2.70757154290335912035174292591, −1.20903961340541875322938612455, 1.12841179400576921434623352143, 2.62577199349411742179891815778, 3.96240444354551282811832752739, 4.69580960775803873789133841995, 5.99269046729600287746644129766, 6.89261007070198055991557205592, 7.47677292364731850174481204849, 8.861310055164679123751109167298, 9.303195903446622637082562170068, 10.25299085848919381388883783340

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