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Properties

Label 2-760-95.74-c1-0-17
Degree $2$
Conductor $760$
Sign $0.999 + 0.0251i$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
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  + (0.248 − 0.295i)3-s + (1.42 − 1.72i)5-s + (1.75 − 1.01i)7-s + (0.495 + 2.80i)9-s + (−2.49 + 4.32i)11-s + (3.32 + 3.95i)13-s + (−0.157 − 0.849i)15-s + (2.75 + 0.485i)17-s + (2.11 + 3.81i)19-s + (0.136 − 0.772i)21-s + (2.36 − 6.50i)23-s + (−0.954 − 4.90i)25-s + (1.95 + 1.12i)27-s + (−0.923 − 5.23i)29-s + (−3.07 − 5.32i)31-s + ⋯
L(s)  = 1  + (0.143 − 0.170i)3-s + (0.636 − 0.771i)5-s + (0.664 − 0.383i)7-s + (0.165 + 0.935i)9-s + (−0.753 + 1.30i)11-s + (0.920 + 1.09i)13-s + (−0.0406 − 0.219i)15-s + (0.668 + 0.117i)17-s + (0.484 + 0.874i)19-s + (0.0297 − 0.168i)21-s + (0.493 − 1.35i)23-s + (−0.190 − 0.981i)25-s + (0.376 + 0.217i)27-s + (−0.171 − 0.972i)29-s + (−0.552 − 0.956i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.999 + 0.0251i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ 0.999 + 0.0251i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97564 - 0.0248024i\)
\(L(\frac12)\) \(\approx\) \(1.97564 - 0.0248024i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.42 + 1.72i)T \)
19 \( 1 + (-2.11 - 3.81i)T \)
good3 \( 1 + (-0.248 + 0.295i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (-1.75 + 1.01i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.49 - 4.32i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.32 - 3.95i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.75 - 0.485i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-2.36 + 6.50i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.923 + 5.23i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (3.07 + 5.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.614iT - 37T^{2} \)
41 \( 1 + (-3.88 - 3.26i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.00 + 2.76i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (10.7 - 1.88i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-4.43 + 12.1i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.214 + 1.21i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-11.9 - 4.36i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.339 - 0.0598i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (12.3 - 4.51i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (2.77 - 3.30i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (1.12 + 0.941i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.12 + 3.53i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.07 - 1.74i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (10.4 + 1.85i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.16455669555200799513549541581, −9.646081804542409492379234899354, −8.428000000081472033328077348380, −7.921377160198417595387900211371, −6.96126868459616490765353742391, −5.76775036750207100019447196693, −4.84156957338393960065844303528, −4.18171103979023194062609163885, −2.26741412056775439844051384836, −1.48061410340277445825479417549, 1.22626403386279071411032343868, 3.03245535824898240287913661595, 3.39749658533985768473456210599, 5.33867798171263884468312241127, 5.68006742200922987628272943844, 6.82500823502295985041188146050, 7.80742129665961956403180911101, 8.725832079888051633076451670314, 9.431658605369946221755992967015, 10.49355710602878034443037670804

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