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Properties

Label 2-76-19.7-c3-0-1
Degree $2$
Conductor $76$
Sign $0.175 - 0.984i$
Analytic cond. $4.48414$
Root an. cond. $2.11758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.70 + 8.15i)3-s + (−1.90 − 3.30i)5-s + 30.9·7-s + (−30.8 + 53.4i)9-s − 43.7·11-s + (−13.5 + 23.3i)13-s + (17.9 − 31.1i)15-s + (−9.75 − 16.9i)17-s + (70.1 − 44.0i)19-s + (145. + 252. i)21-s + (2.21 − 3.84i)23-s + (55.2 − 95.6i)25-s − 327.·27-s + (138. − 240. i)29-s + 137.·31-s + ⋯
L(s)  = 1  + (0.906 + 1.56i)3-s + (−0.170 − 0.295i)5-s + 1.66·7-s + (−1.14 + 1.97i)9-s − 1.19·11-s + (−0.288 + 0.498i)13-s + (0.309 − 0.535i)15-s + (−0.139 − 0.241i)17-s + (0.847 − 0.531i)19-s + (1.51 + 2.62i)21-s + (0.0201 − 0.0348i)23-s + (0.441 − 0.765i)25-s − 2.33·27-s + (0.888 − 1.53i)29-s + 0.796·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.53447 + 1.28540i\)
\(L(\frac12)\) \(\approx\) \(1.53447 + 1.28540i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-70.1 + 44.0i)T \)
good3 \( 1 + (-4.70 - 8.15i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (1.90 + 3.30i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 - 30.9T + 343T^{2} \)
11 \( 1 + 43.7T + 1.33e3T^{2} \)
13 \( 1 + (13.5 - 23.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (9.75 + 16.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-2.21 + 3.84i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-138. + 240. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 137.T + 2.97e4T^{2} \)
37 \( 1 + 232.T + 5.06e4T^{2} \)
41 \( 1 + (-159. - 276. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (116. + 202. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (150. - 259. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-131. + 228. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (277. + 481. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (352. - 611. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-342. + 593. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (182. + 316. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-264. - 458. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-32.4 - 56.1i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.41e3T + 5.71e5T^{2} \)
89 \( 1 + (154. - 267. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-45.7 - 79.3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.34693388358875166483579329686, −13.68143040752012976167554392035, −11.74153465853936204379125732841, −10.76303022512117854902679994129, −9.767061706968984147509183376183, −8.517326257565900505659735627101, −7.86747610001388763264267160020, −5.04700515992034414480932852521, −4.49069427159135054545558019218, −2.60335085547351583937677487003, 1.45402902860194785289541491321, 2.90396659018602866538037346941, 5.29718099302384478545307467655, 7.14906485454892914663439341813, 7.88815698468100688308830453255, 8.632765777540573385151122012094, 10.60628597283235295798351628176, 11.84291607112893274575343304087, 12.77377323112826161649733827234, 13.87570887201809296658698087873

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