Properties

Label 2-76-19.18-c6-0-2
Degree $2$
Conductor $76$
Sign $-0.364 - 0.931i$
Analytic cond. $17.4841$
Root an. cond. $4.18140$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.8i·3-s − 18.1·5-s + 85.5·7-s + 536.·9-s − 316.·11-s + 2.45e3i·13-s − 251. i·15-s − 3.22e3·17-s + (2.50e3 + 6.38e3i)19-s + 1.18e3i·21-s − 7.82e3·23-s − 1.52e4·25-s + 1.75e4i·27-s + 1.89e4i·29-s + 3.23e4i·31-s + ⋯
L(s)  = 1  + 0.514i·3-s − 0.144·5-s + 0.249·7-s + 0.735·9-s − 0.237·11-s + 1.11i·13-s − 0.0744i·15-s − 0.656·17-s + (0.364 + 0.931i)19-s + 0.128i·21-s − 0.643·23-s − 0.978·25-s + 0.892i·27-s + 0.778i·29-s + 1.08i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.364 - 0.931i$
Analytic conductor: \(17.4841\)
Root analytic conductor: \(4.18140\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3),\ -0.364 - 0.931i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.836896 + 1.22692i\)
\(L(\frac12)\) \(\approx\) \(0.836896 + 1.22692i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-2.50e3 - 6.38e3i)T \)
good3 \( 1 - 13.8iT - 729T^{2} \)
5 \( 1 + 18.1T + 1.56e4T^{2} \)
7 \( 1 - 85.5T + 1.17e5T^{2} \)
11 \( 1 + 316.T + 1.77e6T^{2} \)
13 \( 1 - 2.45e3iT - 4.82e6T^{2} \)
17 \( 1 + 3.22e3T + 2.41e7T^{2} \)
23 \( 1 + 7.82e3T + 1.48e8T^{2} \)
29 \( 1 - 1.89e4iT - 5.94e8T^{2} \)
31 \( 1 - 3.23e4iT - 8.87e8T^{2} \)
37 \( 1 - 1.54e4iT - 2.56e9T^{2} \)
41 \( 1 - 9.41e4iT - 4.75e9T^{2} \)
43 \( 1 - 3.92e4T + 6.32e9T^{2} \)
47 \( 1 - 9.23e4T + 1.07e10T^{2} \)
53 \( 1 + 1.30e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.80e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.24e5T + 5.15e10T^{2} \)
67 \( 1 + 1.54e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.77e5iT - 1.28e11T^{2} \)
73 \( 1 - 3.14e5T + 1.51e11T^{2} \)
79 \( 1 + 4.62e5iT - 2.43e11T^{2} \)
83 \( 1 - 6.01e5T + 3.26e11T^{2} \)
89 \( 1 - 2.75e5iT - 4.96e11T^{2} \)
97 \( 1 - 8.52e5iT - 8.32e11T^{2} \)
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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

−13.73565509210489010790066809131, −12.47162382741404136690591056742, −11.38352828330163779999212674280, −10.23730585214808248687528911700, −9.254478835036504914968474841988, −7.890083786603145152147603804538, −6.56865636850394306210888730745, −4.90187647161800223428812063302, −3.76795837058055505795475155777, −1.71789443610224450913029908938, 0.56813098159546206864079020673, 2.30072435624183687155066593109, 4.18133599241282527426623013226, 5.75241330727500396079620830143, 7.21877481102506702533306406393, 8.099649026107955934554860185292, 9.605661706029356167141268217963, 10.78093167145188827221952823243, 11.96875303884928990701082475318, 13.02987405319089056447255992781

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