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Properties

Label 2-28e2-112.109-c1-0-42
Degree $2$
Conductor $784$
Sign $0.757 + 0.652i$
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  + (1.38 − 0.303i)2-s + (−2.07 + 0.554i)3-s + (1.81 − 0.837i)4-s + (−0.992 − 0.265i)5-s + (−2.69 + 1.39i)6-s + (2.25 − 1.70i)8-s + (1.38 − 0.797i)9-s + (−1.45 − 0.0664i)10-s + (−0.340 − 1.27i)11-s + (−3.29 + 2.74i)12-s + (3.98 + 3.98i)13-s + 2.20·15-s + (2.59 − 3.04i)16-s + (2.81 − 4.87i)17-s + (1.66 − 1.52i)18-s + (1.25 − 4.69i)19-s + ⋯
L(s)  = 1  + (0.976 − 0.214i)2-s + (−1.19 + 0.320i)3-s + (0.908 − 0.418i)4-s + (−0.443 − 0.118i)5-s + (−1.09 + 0.569i)6-s + (0.797 − 0.603i)8-s + (0.460 − 0.265i)9-s + (−0.458 − 0.0210i)10-s + (−0.102 − 0.383i)11-s + (−0.951 + 0.791i)12-s + (1.10 + 1.10i)13-s + 0.568·15-s + (0.649 − 0.760i)16-s + (0.682 − 1.18i)17-s + (0.392 − 0.358i)18-s + (0.288 − 1.07i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.757 + 0.652i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.757 + 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73234 - 0.643125i\)
\(L(\frac12)\) \(\approx\) \(1.73234 - 0.643125i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.303i)T \)
7 \( 1 \)
good3 \( 1 + (2.07 - 0.554i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.992 + 0.265i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.340 + 1.27i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.98 - 3.98i)T + 13iT^{2} \)
17 \( 1 + (-2.81 + 4.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.25 + 4.69i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.495 + 0.286i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.81 - 5.81i)T + 29iT^{2} \)
31 \( 1 + (-1.27 + 2.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.52 - 2.01i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.12iT - 41T^{2} \)
43 \( 1 + (-0.783 + 0.783i)T - 43iT^{2} \)
47 \( 1 + (4.77 + 8.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.26 + 4.72i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.71 - 13.8i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.32 - 8.68i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (8.24 - 2.20i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.57iT - 71T^{2} \)
73 \( 1 + (-10.9 - 6.32i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.29 - 7.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.65 + 2.65i)T + 83iT^{2} \)
89 \( 1 + (2.44 - 1.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.98T + 97T^{2} \)
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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

−10.60372801720874276892458039054, −9.647775965555731828521773405254, −8.478926573206237089762543713193, −7.18211227464442760391215121832, −6.45653325941505400566597317379, −5.59341855172180684664791138299, −4.81378723752776392320723302099, −4.02184046504747913948999357752, −2.79156451275614616378497212446, −0.931664035547270351146563930011, 1.33654889553473496011145556331, 3.15274500957412907090301359234, 4.08339967026692133386664469097, 5.23039568294112412797172666746, 6.05366659405342414257449868286, 6.39611567625232998670298893208, 7.86792019411141049521458288904, 8.036347718048807294746938285754, 9.928754344842509798170723481745, 10.79841689531695386314847070004

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