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Properties

Label 2-28e2-112.37-c1-0-14
Degree $2$
Conductor $784$
Sign $-0.994 - 0.108i$
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  + (−0.892 + 1.09i)2-s + (2.20 + 0.589i)3-s + (−0.407 − 1.95i)4-s + (−2.32 + 0.622i)5-s + (−2.60 + 1.88i)6-s + (2.51 + 1.29i)8-s + (1.89 + 1.09i)9-s + (1.38 − 3.10i)10-s + (−0.00762 + 0.0284i)11-s + (0.257 − 4.54i)12-s + (−4.38 + 4.38i)13-s − 5.47·15-s + (−3.66 + 1.59i)16-s + (−1.36 − 2.35i)17-s + (−2.89 + 1.10i)18-s + (1.53 + 5.73i)19-s + ⋯
L(s)  = 1  + (−0.630 + 0.775i)2-s + (1.27 + 0.340i)3-s + (−0.203 − 0.978i)4-s + (−1.03 + 0.278i)5-s + (−1.06 + 0.770i)6-s + (0.888 + 0.459i)8-s + (0.631 + 0.364i)9-s + (0.439 − 0.981i)10-s + (−0.00229 + 0.00857i)11-s + (0.0742 − 1.31i)12-s + (−1.21 + 1.21i)13-s − 1.41·15-s + (−0.916 + 0.399i)16-s + (−0.330 − 0.571i)17-s + (−0.681 + 0.259i)18-s + (0.352 + 1.31i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.994 - 0.108i$
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: \((2,\ 784,\ (\ :1/2),\ -0.994 - 0.108i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0454746 + 0.835134i\)
\(L(\frac12)\) \(\approx\) \(0.0454746 + 0.835134i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.892 - 1.09i)T \)
7 \( 1 \)
good3 \( 1 + (-2.20 - 0.589i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (2.32 - 0.622i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.00762 - 0.0284i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (4.38 - 4.38i)T - 13iT^{2} \)
17 \( 1 + (1.36 + 2.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.53 - 5.73i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.33 - 1.92i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.93 - 4.93i)T - 29iT^{2} \)
31 \( 1 + (-1.29 - 2.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.83 - 2.09i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.207iT - 41T^{2} \)
43 \( 1 + (0.278 + 0.278i)T + 43iT^{2} \)
47 \( 1 + (-1.91 + 3.31i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.328 - 1.22i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.0558 - 0.208i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.59 - 5.93i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-11.9 - 3.20i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.37iT - 71T^{2} \)
73 \( 1 + (3.67 - 2.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.51 - 7.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.55 + 7.55i)T - 83iT^{2} \)
89 \( 1 + (-3.03 - 1.75i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−10.33688143875264223140898278893, −9.511766159420312524160747245881, −8.963212224471947031993921420929, −8.146077722803119731603273899168, −7.36645432339550205656208573036, −6.91009084559834431594810915485, −5.34082127471255798771399468182, −4.29494390433757269033256645365, −3.34186978722156845552162643367, −1.94238261797600858396142331608, 0.42437983309523319282772114492, 2.21929587614653641727409107888, 3.03227317639591832742150967980, 3.94103530045862174544442498173, 5.03459046741615316924917914076, 7.02043787036495069733246913931, 7.71764255422015659492104904080, 8.186029522003671704237630913400, 8.985498583892827136270334993506, 9.680525606807114358197011712384

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