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Properties

Label 2-28e2-112.53-c1-0-61
Degree $2$
Conductor $784$
Sign $0.964 + 0.262i$
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  + (1.36 + 0.366i)2-s + (1.73 + i)4-s + (0.732 − 2.73i)5-s + (1.99 + 2i)8-s + (2.59 − 1.5i)9-s + (2 − 3.46i)10-s + (−1.36 + 0.366i)11-s + (1.99 + 3.46i)16-s + (1 − 1.73i)17-s + (4.09 − 1.09i)18-s + (−2.73 − 0.732i)19-s + (4 − 3.99i)20-s − 2·22-s + (−5.19 + 3i)23-s + (−2.59 − 1.5i)25-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.5i)4-s + (0.327 − 1.22i)5-s + (0.707 + 0.707i)8-s + (0.866 − 0.5i)9-s + (0.632 − 1.09i)10-s + (−0.411 + 0.110i)11-s + (0.499 + 0.866i)16-s + (0.242 − 0.420i)17-s + (0.965 − 0.258i)18-s + (−0.626 − 0.167i)19-s + (0.894 − 0.894i)20-s − 0.426·22-s + (−1.08 + 0.625i)23-s + (−0.519 − 0.300i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.964 + 0.262i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.964 + 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.05221 - 0.408275i\)
\(L(\frac12)\) \(\approx\) \(3.05221 - 0.408275i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
7 \( 1 \)
good3 \( 1 + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.732 + 2.73i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.36 - 0.366i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.73 + 0.732i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7 + 7i)T - 29iT^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.83 - 6.83i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + (1 + i)T + 43iT^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.36 + 0.366i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (10.9 - 2.92i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (8.19 + 2.19i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.09 - 4.09i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5.19 - 3i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10 - 10i)T - 83iT^{2} \)
89 \( 1 + (12.1 - 7i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.13696508448401606334250718662, −9.533051054438390270995368320578, −8.303487714561042492317346122133, −7.69847863376591921030317397108, −6.48946342062086597994367102176, −5.80280881340403431561832012204, −4.63461907771490440489240842704, −4.27247825105335320822774468829, −2.74012109803382867697427536705, −1.35305072673677483197698843644, 1.82467066672847921935139719877, 2.78889154640730714416376129004, 3.84051169651873889633789757093, 4.86697436391067708894425446437, 5.92520273227399064649417564804, 6.75826527743788549346270447167, 7.34871126530700466092988041749, 8.534140851428488681479736752943, 10.10983241351852536918218731449, 10.47518728841281710974353930762

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