Properties

Label 2-28e2-112.109-c1-0-72
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

\(L(1)\) \(\approx\) \(3.86471 - 1.43476i\)
\(L(\frac12)\) \(\approx\) \(3.86471 - 1.43476i\)
\(L(\frac{3}{2})\) 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

−10.31045516025883586740350227760, −9.455189238990190294365736080874, −8.206045797230451050471755786108, −7.77426270473196036024194409927, −6.55308275166769349893536232878, −5.77237378188493841063640116250, −4.65866765417401896516626274917, −3.45149074340568411643529389875, −2.68778851847386506286903251538, −1.73903755255325933565755507800, 2.31011143749517332209666192964, 2.63950200377390653579518149418, 4.17306875347413676478798947335, 4.67007659466170164492819051567, 5.89389034435361632501407434354, 7.05081516899288673859596345969, 7.56203415615003686642894491120, 8.833917183116002010835028862889, 9.377300024744858355415937333743, 10.25186484639495934126801055567

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