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Properties

Label 2-28e2-112.109-c1-0-49
Degree $2$
Conductor $784$
Sign $-0.849 + 0.526i$
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  + (−0.146 − 1.40i)2-s + (−0.977 + 0.261i)3-s + (−1.95 + 0.412i)4-s + (1.18 + 0.317i)5-s + (0.511 + 1.33i)6-s + (0.867 + 2.69i)8-s + (−1.71 + 0.988i)9-s + (0.272 − 1.71i)10-s + (−1.08 − 4.06i)11-s + (1.80 − 0.915i)12-s + (2.02 + 2.02i)13-s − 1.24·15-s + (3.65 − 1.61i)16-s + (0.132 − 0.229i)17-s + (1.64 + 2.26i)18-s + (1.66 − 6.20i)19-s + ⋯
L(s)  = 1  + (−0.103 − 0.994i)2-s + (−0.564 + 0.151i)3-s + (−0.978 + 0.206i)4-s + (0.530 + 0.142i)5-s + (0.208 + 0.545i)6-s + (0.306 + 0.951i)8-s + (−0.570 + 0.329i)9-s + (0.0862 − 0.541i)10-s + (−0.328 − 1.22i)11-s + (0.520 − 0.264i)12-s + (0.560 + 0.560i)13-s − 0.320·15-s + (0.914 − 0.403i)16-s + (0.0320 − 0.0555i)17-s + (0.386 + 0.533i)18-s + (0.381 − 1.42i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.849 + 0.526i$
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: \((2,\ 784,\ (\ :1/2),\ -0.849 + 0.526i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.214293 - 0.752375i\)
\(L(\frac12)\) \(\approx\) \(0.214293 - 0.752375i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.146 + 1.40i)T \)
7 \( 1 \)
good3 \( 1 + (0.977 - 0.261i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-1.18 - 0.317i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.08 + 4.06i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.02 - 2.02i)T + 13iT^{2} \)
17 \( 1 + (-0.132 + 0.229i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.66 + 6.20i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.33 - 0.773i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.328 + 0.328i)T + 29iT^{2} \)
31 \( 1 + (-3.02 + 5.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.08 + 2.43i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 11.0iT - 41T^{2} \)
43 \( 1 + (-3.38 + 3.38i)T - 43iT^{2} \)
47 \( 1 + (-1.56 - 2.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.157 - 0.588i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.69 - 6.31i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.78 - 6.64i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (4.56 - 1.22i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.03iT - 71T^{2} \)
73 \( 1 + (12.8 + 7.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.29 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.715 - 0.715i)T + 83iT^{2} \)
89 \( 1 + (9.51 - 5.49i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−10.30382708557652608080955171394, −9.059581867620063219883634910547, −8.691612240831059799939730783500, −7.48877557372438277976897214055, −6.03207821983756193029480266779, −5.50117771139703820634146351256, −4.39214352799687539456088950468, −3.18867377695684294561225435054, −2.16864011178727554546666097605, −0.46067352066132051171115427465, 1.44557125903602089891961907465, 3.39304603026525913080328231747, 4.71897590997369392434052563706, 5.58195778203699731517634250951, 6.15341852484640392480846494274, 7.08325290787290626891347819175, 8.025824935346743272520144102983, 8.768925494418879896450264207375, 9.912046271539203444221031257868, 10.20513247292027284781138701279

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