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Properties

Label 2-28e2-112.53-c1-0-9
Degree $2$
Conductor $784$
Sign $-0.317 + 0.948i$
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  + (0.164 + 1.40i)2-s + (0.338 + 1.26i)3-s + (−1.94 + 0.461i)4-s + (−0.958 + 3.57i)5-s + (−1.72 + 0.683i)6-s + (−0.968 − 2.65i)8-s + (1.11 − 0.642i)9-s + (−5.17 − 0.757i)10-s + (−4.80 + 1.28i)11-s + (−1.24 − 2.30i)12-s + (−3.14 + 3.14i)13-s − 4.84·15-s + (3.57 − 1.79i)16-s + (3.33 − 5.76i)17-s + (1.08 + 1.45i)18-s + (4.73 + 1.26i)19-s + ⋯
L(s)  = 1  + (0.116 + 0.993i)2-s + (0.195 + 0.730i)3-s + (−0.972 + 0.230i)4-s + (−0.428 + 1.59i)5-s + (−0.702 + 0.279i)6-s + (−0.342 − 0.939i)8-s + (0.371 − 0.214i)9-s + (−1.63 − 0.239i)10-s + (−1.44 + 0.388i)11-s + (−0.358 − 0.665i)12-s + (−0.871 + 0.871i)13-s − 1.25·15-s + (0.893 − 0.449i)16-s + (0.807 − 1.39i)17-s + (0.256 + 0.343i)18-s + (1.08 + 0.291i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.317 + 0.948i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.317 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534809 - 0.743356i\)
\(L(\frac12)\) \(\approx\) \(0.534809 - 0.743356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.164 - 1.40i)T \)
7 \( 1 \)
good3 \( 1 + (-0.338 - 1.26i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.958 - 3.57i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (4.80 - 1.28i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.14 - 3.14i)T - 13iT^{2} \)
17 \( 1 + (-3.33 + 5.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.73 - 1.26i)T + (16.4 + 9.5i)T^{2} \)
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 \( 1 + (-0.343 + 1.28i)T + (-32.0 - 18.5i)T^{2} \)
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 \( 1 + (-1.83 + 0.490i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.24 - 1.40i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.52 - 1.47i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.83 + 10.5i)T + (-58.0 + 33.5i)T^{2} \)
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

−10.54845793956099284446656078781, −9.849737481507675678263193952962, −9.452794874493149177880219988775, −7.951866688798340932336754579830, −7.28465978625339765272901008509, −6.91209154004507472095000378219, −5.51126699257653005119623462924, −4.71260135988756824763128803498, −3.58987066364856131589496400391, −2.79552893718686753453987024866, 0.44267490794166855234971960443, 1.59409359018406151688053730223, 2.84325492692602823842444095061, 4.14461193638948553437981728034, 5.12980738050683857526773862336, 5.67112899047158226391438449306, 7.76432191037997210949250799410, 7.910200943573943718231484365673, 8.771354171028289698637287493015, 9.921435591545091122043896555682

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