Properties

Label 2-76-4.3-c2-0-8
Degree $2$
Conductor $76$
Sign $0.436 - 0.899i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.94 + 0.447i)2-s + 3.19i·3-s + (3.59 + 1.74i)4-s − 3.90·5-s + (−1.43 + 6.23i)6-s − 2.64i·7-s + (6.23 + 5.01i)8-s − 1.23·9-s + (−7.60 − 1.74i)10-s − 13.2i·11-s + (−5.58 + 11.5i)12-s + 1.60·13-s + (1.18 − 5.15i)14-s − 12.4i·15-s + (9.91 + 12.5i)16-s + 8.10·17-s + ⋯
L(s)  = 1  + (0.974 + 0.223i)2-s + 1.06i·3-s + (0.899 + 0.436i)4-s − 0.780·5-s + (−0.238 + 1.03i)6-s − 0.378i·7-s + (0.779 + 0.626i)8-s − 0.137·9-s + (−0.760 − 0.174i)10-s − 1.20i·11-s + (−0.465 + 0.959i)12-s + 0.123·13-s + (0.0845 − 0.368i)14-s − 0.832i·15-s + (0.619 + 0.784i)16-s + 0.476·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.436 - 0.899i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.436 - 0.899i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.67127 + 1.04718i\)
\(L(\frac12)\) \(\approx\) \(1.67127 + 1.04718i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.94 - 0.447i)T \)
19 \( 1 - 4.35iT \)
good3 \( 1 - 3.19iT - 9T^{2} \)
5 \( 1 + 3.90T + 25T^{2} \)
7 \( 1 + 2.64iT - 49T^{2} \)
11 \( 1 + 13.2iT - 121T^{2} \)
13 \( 1 - 1.60T + 169T^{2} \)
17 \( 1 - 8.10T + 289T^{2} \)
23 \( 1 + 38.2iT - 529T^{2} \)
29 \( 1 + 51.1T + 841T^{2} \)
31 \( 1 - 13.6iT - 961T^{2} \)
37 \( 1 + 35.3T + 1.36e3T^{2} \)
41 \( 1 - 8.06T + 1.68e3T^{2} \)
43 \( 1 - 16.2iT - 1.84e3T^{2} \)
47 \( 1 - 1.59iT - 2.20e3T^{2} \)
53 \( 1 - 56.8T + 2.80e3T^{2} \)
59 \( 1 - 106. iT - 3.48e3T^{2} \)
61 \( 1 + 35.9T + 3.72e3T^{2} \)
67 \( 1 - 47.5iT - 4.48e3T^{2} \)
71 \( 1 + 125. iT - 5.04e3T^{2} \)
73 \( 1 - 34.2T + 5.32e3T^{2} \)
79 \( 1 - 71.1iT - 6.24e3T^{2} \)
83 \( 1 + 149. iT - 6.88e3T^{2} \)
89 \( 1 + 169.T + 7.92e3T^{2} \)
97 \( 1 - 109.T + 9.40e3T^{2} \)
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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

−14.61817478156520369964272926220, −13.55144010006253019445719085047, −12.29578474942570283597827426532, −11.15758244361419508997785488936, −10.39840963082615081694052426746, −8.631660007393844863658705836710, −7.33525850884920181571913644323, −5.73038517966097620047823738092, −4.31427087735724631054194832323, −3.43291263748503298481147151087, 1.89647913782104993580287036651, 3.86245796768298382754820290981, 5.51882181003969479708021739931, 7.06172075844459048601310830793, 7.69718132623534560604355200338, 9.736925690919939146036487800368, 11.34532093716463591474445593196, 12.13445015173975691464552008637, 12.85783837680268847019466082082, 13.81487907523145300141753307341

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