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Properties

Label 2-28e2-112.53-c1-0-60
Degree $2$
Conductor $784$
Sign $-0.957 + 0.289i$
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  + (−0.329 − 1.37i)2-s + (−0.231 − 0.862i)3-s + (−1.78 + 0.905i)4-s + (0.857 − 3.20i)5-s + (−1.10 + 0.601i)6-s + (1.83 + 2.15i)8-s + (1.90 − 1.10i)9-s + (−4.68 − 0.125i)10-s + (2.98 − 0.799i)11-s + (1.19 + 1.32i)12-s + (4.03 − 4.03i)13-s − 2.95·15-s + (2.35 − 3.22i)16-s + (0.173 − 0.301i)17-s + (−2.14 − 2.26i)18-s + (5.82 + 1.56i)19-s + ⋯
L(s)  = 1  + (−0.232 − 0.972i)2-s + (−0.133 − 0.497i)3-s + (−0.891 + 0.452i)4-s + (0.383 − 1.43i)5-s + (−0.453 + 0.245i)6-s + (0.647 + 0.761i)8-s + (0.636 − 0.367i)9-s + (−1.48 − 0.0397i)10-s + (0.899 − 0.240i)11-s + (0.344 + 0.383i)12-s + (1.11 − 1.11i)13-s − 0.763·15-s + (0.589 − 0.807i)16-s + (0.0421 − 0.0730i)17-s + (−0.505 − 0.533i)18-s + (1.33 + 0.357i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.957 + 0.289i$
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: \((2,\ 784,\ (\ :1/2),\ -0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.216805 - 1.46641i\)
\(L(\frac12)\) \(\approx\) \(0.216805 - 1.46641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.329 + 1.37i)T \)
7 \( 1 \)
good3 \( 1 + (0.231 + 0.862i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.857 + 3.20i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.98 + 0.799i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-4.03 + 4.03i)T - 13iT^{2} \)
17 \( 1 + (-0.173 + 0.301i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.82 - 1.56i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.40 - 3.11i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.21 + 1.21i)T - 29iT^{2} \)
31 \( 1 + (0.631 - 1.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.35 - 8.77i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.68iT - 41T^{2} \)
43 \( 1 + (-4.05 - 4.05i)T + 43iT^{2} \)
47 \( 1 + (-2.32 - 4.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.5 - 3.08i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (7.07 - 1.89i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.00728 + 0.00195i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.10 - 4.12i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.828iT - 71T^{2} \)
73 \( 1 + (-5.41 - 3.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.377 - 0.654i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.66 + 3.66i)T - 83iT^{2} \)
89 \( 1 + (-5.40 + 3.12i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.18T + 97T^{2} \)
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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

−9.700955501629173500632089596169, −9.319524764547691889376561704684, −8.314134984520442448783966142383, −7.72830590048047039894642688114, −6.18845363608630915510444196145, −5.35460076544311508378504398338, −4.24510424520963249200712512038, −3.32459046931741817895711812168, −1.46043781666213763157298729467, −1.03320179541874854878556323304, 1.75502756683629131862877016422, 3.59898240660010026729410736209, 4.34659869161791663623500787990, 5.61889956600002812997888346364, 6.53581279308796838552330469749, 6.97957489417987506723189090615, 7.928854082294946302334913724769, 9.218918331427728869444097398752, 9.624201485911175601928715562842, 10.58579468535542518756949903601

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