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Properties

Label 2-28e2-112.109-c1-0-19
Degree $2$
Conductor $784$
Sign $-0.968 - 0.247i$
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  + (1.18 + 0.765i)2-s + (−3.19 + 0.856i)3-s + (0.829 + 1.82i)4-s + (0.890 + 0.238i)5-s + (−4.45 − 1.42i)6-s + (−0.406 + 2.79i)8-s + (6.87 − 3.97i)9-s + (0.876 + 0.964i)10-s + (0.873 + 3.26i)11-s + (−4.20 − 5.10i)12-s + (3.39 + 3.39i)13-s − 3.04·15-s + (−2.62 + 3.01i)16-s + (−1.40 + 2.43i)17-s + (11.2 + 0.539i)18-s + (0.392 − 1.46i)19-s + ⋯
L(s)  = 1  + (0.841 + 0.541i)2-s + (−1.84 + 0.494i)3-s + (0.414 + 0.910i)4-s + (0.398 + 0.106i)5-s + (−1.81 − 0.582i)6-s + (−0.143 + 0.989i)8-s + (2.29 − 1.32i)9-s + (0.277 + 0.305i)10-s + (0.263 + 0.983i)11-s + (−1.21 − 1.47i)12-s + (0.941 + 0.941i)13-s − 0.787·15-s + (−0.656 + 0.754i)16-s + (−0.341 + 0.591i)17-s + (2.64 + 0.127i)18-s + (0.0901 − 0.336i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.968 - 0.247i$
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: \((2,\ 784,\ (\ :1/2),\ -0.968 - 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159764 + 1.26909i\)
\(L(\frac12)\) \(\approx\) \(0.159764 + 1.26909i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 - 0.765i)T \)
7 \( 1 \)
good3 \( 1 + (3.19 - 0.856i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.890 - 0.238i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.873 - 3.26i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.39 - 3.39i)T + 13iT^{2} \)
17 \( 1 + (1.40 - 2.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.392 + 1.46i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.94 - 2.27i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.76 + 5.76i)T + 29iT^{2} \)
31 \( 1 + (-0.611 + 1.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.40 - 2.51i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.10iT - 41T^{2} \)
43 \( 1 + (2.53 - 2.53i)T - 43iT^{2} \)
47 \( 1 + (1.77 + 3.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.48 - 5.54i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.724 - 2.70i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.53 - 13.1i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.30 - 0.349i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.695iT - 71T^{2} \)
73 \( 1 + (-0.275 - 0.159i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.32 + 2.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.22 - 6.22i)T + 83iT^{2} \)
89 \( 1 + (5.18 - 2.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−10.94893054092628443527472652961, −10.01136737144438497084431162406, −9.204541936873559000008803544533, −7.69487430100121004028507666818, −6.72169821774216522325962712591, −6.12837002956985520794275474264, −5.56905441867935996363456982782, −4.29669817248553978149897684809, −4.09694959297213279791371499525, −1.85760346385752627760665417901, 0.62740692300220609978871341577, 1.75656639811728747137842189076, 3.49628595743985235373874063098, 4.69168624181270909426751427169, 5.67288582608785637327050994184, 5.95437393085890267210403236326, 6.77674747647037556620403992287, 7.934385688379023103283298218924, 9.483959118924372690975810694598, 10.36085062797499593285879790568

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