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Properties

Label 2-28e2-112.109-c1-0-66
Degree $2$
Conductor $784$
Sign $0.115 + 0.993i$
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  + (0.533 + 1.30i)2-s + (1.65 − 0.442i)3-s + (−1.43 + 1.39i)4-s + (−3.54 − 0.949i)5-s + (1.45 + 1.92i)6-s + (−2.59 − 1.12i)8-s + (−0.0695 + 0.0401i)9-s + (−0.647 − 5.14i)10-s + (−0.395 − 1.47i)11-s + (−1.74 + 2.93i)12-s + (−2.97 − 2.97i)13-s − 6.26·15-s + (0.0933 − 3.99i)16-s + (3.59 − 6.22i)17-s + (−0.0896 − 0.0696i)18-s + (0.826 − 3.08i)19-s + ⋯
L(s)  = 1  + (0.377 + 0.926i)2-s + (0.952 − 0.255i)3-s + (−0.715 + 0.698i)4-s + (−1.58 − 0.424i)5-s + (0.595 + 0.786i)6-s + (−0.917 − 0.398i)8-s + (−0.0231 + 0.0133i)9-s + (−0.204 − 1.62i)10-s + (−0.119 − 0.444i)11-s + (−0.503 + 0.848i)12-s + (−0.826 − 0.826i)13-s − 1.61·15-s + (0.0233 − 0.999i)16-s + (0.871 − 1.50i)17-s + (−0.0211 − 0.0164i)18-s + (0.189 − 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.115 + 0.993i$
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: \((2,\ 784,\ (\ :1/2),\ 0.115 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.541537 - 0.482166i\)
\(L(\frac12)\) \(\approx\) \(0.541537 - 0.482166i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.533 - 1.30i)T \)
7 \( 1 \)
good3 \( 1 + (-1.65 + 0.442i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (3.54 + 0.949i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.395 + 1.47i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.97 + 2.97i)T + 13iT^{2} \)
17 \( 1 + (-3.59 + 6.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.826 + 3.08i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.02 - 1.74i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.851 - 0.851i)T + 29iT^{2} \)
31 \( 1 + (1.97 - 3.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.10 + 2.17i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.67iT - 41T^{2} \)
43 \( 1 + (4.25 - 4.25i)T - 43iT^{2} \)
47 \( 1 + (-1.17 - 2.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.25 + 4.66i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.41 + 5.27i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.514 - 1.92i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (7.79 - 2.08i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (2.88 + 1.66i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.90 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.20 + 1.20i)T + 83iT^{2} \)
89 \( 1 + (10.9 - 6.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.08T + 97T^{2} \)
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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

−9.697293712936524920629243242046, −8.822145188346793709039401975361, −8.169427280334304146017229790767, −7.55911111172612606464275097946, −7.10049754135510408236032481029, −5.42809178612007994381373221273, −4.79090109621746440667800719862, −3.48546350616532617780807420126, −2.98427002980005918344562125984, −0.27765978605720780918795264830, 1.97401477191300654976668224326, 3.20871261445534585632846770216, 3.83333521333859802709273807599, 4.52048674391627931235181473933, 5.95854522611187615728236595207, 7.30041882325422708040355926431, 8.156003794342793817249181575387, 8.748062004139954251831860992918, 9.880209277852637762131814664077, 10.40079137623077401872147768515

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