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Properties

Label 2-28e2-112.93-c1-0-4
Degree $2$
Conductor $784$
Sign $-0.317 - 0.948i$
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  + (0.164 − 1.40i)2-s + (−0.338 + 1.26i)3-s + (−1.94 − 0.461i)4-s + (0.958 + 3.57i)5-s + (1.72 + 0.683i)6-s + (−0.968 + 2.65i)8-s + (1.11 + 0.642i)9-s + (5.17 − 0.757i)10-s + (−4.80 − 1.28i)11-s + (1.24 − 2.30i)12-s + (3.14 + 3.14i)13-s − 4.84·15-s + (3.57 + 1.79i)16-s + (−3.33 − 5.76i)17-s + (1.08 − 1.45i)18-s + (−4.73 + 1.26i)19-s + ⋯
L(s)  = 1  + (0.116 − 0.993i)2-s + (−0.195 + 0.730i)3-s + (−0.972 − 0.230i)4-s + (0.428 + 1.59i)5-s + (0.702 + 0.279i)6-s + (−0.342 + 0.939i)8-s + (0.371 + 0.214i)9-s + (1.63 − 0.239i)10-s + (−1.44 − 0.388i)11-s + (0.358 − 0.665i)12-s + (0.871 + 0.871i)13-s − 1.25·15-s + (0.893 + 0.449i)16-s + (−0.807 − 1.39i)17-s + (0.256 − 0.343i)18-s + (−1.08 + 0.291i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.317 - 0.948i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.317 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.533854 + 0.742029i\)
\(L(\frac12)\) \(\approx\) \(0.533854 + 0.742029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.164 + 1.40i)T \)
7 \( 1 \)
good3 \( 1 + (0.338 - 1.26i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.958 - 3.57i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (4.80 + 1.28i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.14 - 3.14i)T + 13iT^{2} \)
17 \( 1 + (3.33 + 5.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.73 - 1.26i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.83 + 1.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0287 + 0.0287i)T + 29iT^{2} \)
31 \( 1 + (-2.29 - 3.97i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.343 - 1.28i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.49iT - 41T^{2} \)
43 \( 1 + (0.947 - 0.947i)T - 43iT^{2} \)
47 \( 1 + (-0.399 + 0.691i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.83 - 0.490i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.24 - 1.40i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.52 - 1.47i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.83 - 10.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.16iT - 71T^{2} \)
73 \( 1 + (1.13 - 0.654i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.166 - 0.287i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.9 - 10.9i)T + 83iT^{2} \)
89 \( 1 + (-5.63 - 3.25i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.56T + 97T^{2} \)
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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

−10.60767916094255058815504735849, −10.11629837884717531706913127039, −9.237361895587245653419293116302, −8.185910543930400294442108127545, −6.97340080051241831089349183386, −6.02174174685319672131463425509, −4.94012755047491270144830468935, −4.01738738354485593430483426715, −2.92202345525662109506065483611, −2.13228120557296723554934971406, 0.43972369373390238968722834329, 1.88259420780017046313487788344, 3.99060133497389614175294516986, 4.83056293191055037878600439875, 5.80381505500826608795909717053, 6.30093924231266382224594167023, 7.60826888620540714961468060317, 8.234852162655366243827510719857, 8.796293723451518175468537191975, 9.854029224286122264512816679675

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