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Properties

Label 2-28e2-112.37-c1-0-72
Degree $2$
Conductor $784$
Sign $-0.545 + 0.838i$
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  + (1.13 − 0.844i)2-s + (1.26 + 0.338i)3-s + (0.572 − 1.91i)4-s + (−3.57 + 0.958i)5-s + (1.72 − 0.683i)6-s + (−0.968 − 2.65i)8-s + (−1.11 − 0.642i)9-s + (−3.24 + 4.10i)10-s + (1.28 − 4.80i)11-s + (1.37 − 2.22i)12-s + (3.14 − 3.14i)13-s − 4.84·15-s + (−3.34 − 2.19i)16-s + (−3.33 − 5.76i)17-s + (−1.80 + 0.211i)18-s + (1.26 + 4.73i)19-s + ⋯
L(s)  = 1  + (0.802 − 0.597i)2-s + (0.730 + 0.195i)3-s + (0.286 − 0.958i)4-s + (−1.59 + 0.428i)5-s + (0.702 − 0.279i)6-s + (−0.342 − 0.939i)8-s + (−0.371 − 0.214i)9-s + (−1.02 + 1.29i)10-s + (0.388 − 1.44i)11-s + (0.396 − 0.643i)12-s + (0.871 − 0.871i)13-s − 1.25·15-s + (−0.835 − 0.548i)16-s + (−0.807 − 1.39i)17-s + (−0.425 + 0.0498i)18-s + (0.291 + 1.08i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.545 + 0.838i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.545 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.942059 - 1.73715i\)
\(L(\frac12)\) \(\approx\) \(0.942059 - 1.73715i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 + 0.844i)T \)
7 \( 1 \)
good3 \( 1 + (-1.26 - 0.338i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (3.57 - 0.958i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.28 + 4.80i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.14 + 3.14i)T - 13iT^{2} \)
17 \( 1 + (3.33 + 5.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.26 - 4.73i)T + (-16.4 + 9.5i)T^{2} \)
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 \( 1 + (1.28 - 0.343i)T + (32.0 - 18.5i)T^{2} \)
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 \( 1 + (0.490 - 1.83i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.40 - 5.24i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.47 - 5.52i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-10.5 - 2.83i)T + (58.0 + 33.5i)T^{2} \)
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

−10.28352788498466181202799228586, −9.000603168174422361558810147124, −8.443864590976633340696900522086, −7.45782224284567561177855541301, −6.39594230026908841521687681264, −5.38662299126198730133802465938, −4.05793521627742301189064005141, −3.36385616002864206862471961435, −2.93434058521961085026202859562, −0.70634217901062947577959874066, 2.10336174414103081377926940932, 3.48144448878922218910243075974, 4.23505302072993588813177284743, 4.87361825233792492081064040299, 6.46279139685003052419967752844, 7.14028739579104766805764324752, 8.012535501454249667847141883557, 8.548529551995068123674697086203, 9.266579016900935337137529749705, 11.10962716558666917869245776764

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