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Properties

Label 2-28e2-112.37-c1-0-37
Degree $2$
Conductor $784$
Sign $0.998 + 0.0533i$
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  + (−1.40 + 0.164i)2-s + (0.839 + 0.224i)3-s + (1.94 − 0.461i)4-s + (3.16 − 0.847i)5-s + (−1.21 − 0.177i)6-s + (−2.65 + 0.968i)8-s + (−1.94 − 1.12i)9-s + (−4.30 + 1.71i)10-s + (−0.769 + 2.87i)11-s + (1.73 + 0.0499i)12-s + (3.63 − 3.63i)13-s + 2.84·15-s + (3.57 − 1.79i)16-s + (1.81 + 3.14i)17-s + (2.91 + 1.25i)18-s + (0.429 + 1.60i)19-s + ⋯
L(s)  = 1  + (−0.993 + 0.116i)2-s + (0.484 + 0.129i)3-s + (0.972 − 0.230i)4-s + (1.41 − 0.379i)5-s + (−0.496 − 0.0726i)6-s + (−0.939 + 0.342i)8-s + (−0.648 − 0.374i)9-s + (−1.36 + 0.540i)10-s + (−0.231 + 0.865i)11-s + (0.501 + 0.0144i)12-s + (1.00 − 1.00i)13-s + 0.734·15-s + (0.893 − 0.449i)16-s + (0.441 + 0.763i)17-s + (0.687 + 0.296i)18-s + (0.0985 + 0.367i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.998 + 0.0533i$
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: \((2,\ 784,\ (\ :1/2),\ 0.998 + 0.0533i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51180 - 0.0403558i\)
\(L(\frac12)\) \(\approx\) \(1.51180 - 0.0403558i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.164i)T \)
7 \( 1 \)
good3 \( 1 + (-0.839 - 0.224i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-3.16 + 0.847i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.769 - 2.87i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.63 + 3.63i)T - 13iT^{2} \)
17 \( 1 + (-1.81 - 3.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.429 - 1.60i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.33 - 3.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.10 + 5.10i)T - 29iT^{2} \)
31 \( 1 + (-1.00 - 1.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.57 - 1.49i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 3.71iT - 41T^{2} \)
43 \( 1 + (2.91 + 2.91i)T + 43iT^{2} \)
47 \( 1 + (-5.06 + 8.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.986 - 3.68i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.977 - 3.64i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.75 + 6.54i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-5.88 - 1.57i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.55iT - 71T^{2} \)
73 \( 1 + (0.989 - 0.571i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.120 + 0.209i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.459 + 0.459i)T - 83iT^{2} \)
89 \( 1 + (-3.76 - 2.17i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.80T + 97T^{2} \)
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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

−10.14234014826256559507831540039, −9.408979725547003549584874159190, −8.655745657840207402941801627737, −8.081687287087274729640870919241, −6.86070819171613983322427073974, −5.89386109509256155685210824964, −5.36127612366763226241787567424, −3.44085775121653025108336292215, −2.35451661170270752661859777580, −1.22506030461792275883992407835, 1.32074341949699460397479252502, 2.53901028071367153250983068515, 3.18815643508192791631046764270, 5.21486849209667553804290345377, 6.20674419102129205126847364287, 6.80315057291608796056839408885, 7.966015414393637100196838989207, 8.945743770979027092451324294576, 9.134414943458748721960557039347, 10.25931730766111799344062415791

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