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Properties

Label 2-76-76.75-c3-0-25
Degree $2$
Conductor $76$
Sign $0.544 + 0.838i$
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  + (2.25 − 1.71i)2-s + 8.24·3-s + (2.15 − 7.70i)4-s − 10.8·5-s + (18.5 − 14.0i)6-s − 8.32i·7-s + (−8.32 − 21.0i)8-s + 40.9·9-s + (−24.4 + 18.5i)10-s + 53.9i·11-s + (17.7 − 63.5i)12-s + 43.7i·13-s + (−14.2 − 18.7i)14-s − 89.3·15-s + (−54.7 − 33.1i)16-s + 91.7·17-s + ⋯
L(s)  = 1  + (0.796 − 0.604i)2-s + 1.58·3-s + (0.268 − 0.963i)4-s − 0.968·5-s + (1.26 − 0.959i)6-s − 0.449i·7-s + (−0.368 − 0.929i)8-s + 1.51·9-s + (−0.771 + 0.585i)10-s + 1.47i·11-s + (0.426 − 1.52i)12-s + 0.933i·13-s + (−0.271 − 0.358i)14-s − 1.53·15-s + (−0.855 − 0.518i)16-s + 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.544 + 0.838i$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ 0.544 + 0.838i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.65814 - 1.44284i\)
\(L(\frac12)\) \(\approx\) \(2.65814 - 1.44284i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.25 + 1.71i)T \)
19 \( 1 + (54.7 - 62.1i)T \)
good3 \( 1 - 8.24T + 27T^{2} \)
5 \( 1 + 10.8T + 125T^{2} \)
7 \( 1 + 8.32iT - 343T^{2} \)
11 \( 1 - 53.9iT - 1.33e3T^{2} \)
13 \( 1 - 43.7iT - 2.19e3T^{2} \)
17 \( 1 - 91.7T + 4.91e3T^{2} \)
23 \( 1 + 172. iT - 1.21e4T^{2} \)
29 \( 1 - 3.12iT - 2.43e4T^{2} \)
31 \( 1 + 113.T + 2.97e4T^{2} \)
37 \( 1 - 159. iT - 5.06e4T^{2} \)
41 \( 1 + 205. iT - 6.89e4T^{2} \)
43 \( 1 - 28.4iT - 7.95e4T^{2} \)
47 \( 1 - 62.4iT - 1.03e5T^{2} \)
53 \( 1 + 650. iT - 1.48e5T^{2} \)
59 \( 1 + 792.T + 2.05e5T^{2} \)
61 \( 1 - 30.4T + 2.26e5T^{2} \)
67 \( 1 - 278.T + 3.00e5T^{2} \)
71 \( 1 - 74.0T + 3.57e5T^{2} \)
73 \( 1 - 1.11e3T + 3.89e5T^{2} \)
79 \( 1 + 51.9T + 4.93e5T^{2} \)
83 \( 1 + 1.13e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.03e3iT - 7.04e5T^{2} \)
97 \( 1 - 145. iT - 9.12e5T^{2} \)
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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

−14.06997414338319921562715527558, −12.73663720468644611954169416662, −12.04045910431332920330873179094, −10.42979765161811589572957504097, −9.457760010342785522722073681621, −8.024291169362114248554363945694, −6.94577106684527074331096035381, −4.45217412362459865776552343637, −3.62436754376520834205333174710, −2.01503668143485828644869145689, 2.99140025855042977165783126625, 3.74038478083741976464908269487, 5.65324180614317985445806794615, 7.57465248559177404667784651059, 8.125315881409301506018811200613, 9.121270945611848344689902664088, 11.13776807401942272431008674927, 12.38300900720166803943150687011, 13.44395743096047722377679278109, 14.21145547877038364181497832860

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