Properties

Label 2-76-4.3-c2-0-4
Degree $2$
Conductor $76$
Sign $-0.611 - 0.791i$
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  + (0.645 + 1.89i)2-s + 0.820i·3-s + (−3.16 + 2.44i)4-s − 2.38·5-s + (−1.55 + 0.529i)6-s + 12.3i·7-s + (−6.67 − 4.41i)8-s + 8.32·9-s + (−1.53 − 4.50i)10-s − 9.15i·11-s + (−2.00 − 2.59i)12-s + 0.940·13-s + (−23.4 + 7.98i)14-s − 1.95i·15-s + (4.05 − 15.4i)16-s + 27.1·17-s + ⋯
L(s)  = 1  + (0.322 + 0.946i)2-s + 0.273i·3-s + (−0.791 + 0.611i)4-s − 0.476·5-s + (−0.258 + 0.0882i)6-s + 1.76i·7-s + (−0.833 − 0.552i)8-s + 0.925·9-s + (−0.153 − 0.450i)10-s − 0.831i·11-s + (−0.167 − 0.216i)12-s + 0.0723·13-s + (−1.67 + 0.570i)14-s − 0.130i·15-s + (0.253 − 0.967i)16-s + 1.59·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.611 - 0.791i$
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: \((2,\ 76,\ (\ :1),\ -0.611 - 0.791i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.564774 + 1.14936i\)
\(L(\frac12)\) \(\approx\) \(0.564774 + 1.14936i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.645 - 1.89i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 - 0.820iT - 9T^{2} \)
5 \( 1 + 2.38T + 25T^{2} \)
7 \( 1 - 12.3iT - 49T^{2} \)
11 \( 1 + 9.15iT - 121T^{2} \)
13 \( 1 - 0.940T + 169T^{2} \)
17 \( 1 - 27.1T + 289T^{2} \)
23 \( 1 - 13.8iT - 529T^{2} \)
29 \( 1 - 49.8T + 841T^{2} \)
31 \( 1 + 24.6iT - 961T^{2} \)
37 \( 1 + 27.3T + 1.36e3T^{2} \)
41 \( 1 + 38.4T + 1.68e3T^{2} \)
43 \( 1 + 41.2iT - 1.84e3T^{2} \)
47 \( 1 - 45.1iT - 2.20e3T^{2} \)
53 \( 1 + 19.9T + 2.80e3T^{2} \)
59 \( 1 + 34.7iT - 3.48e3T^{2} \)
61 \( 1 - 33.2T + 3.72e3T^{2} \)
67 \( 1 - 3.48iT - 4.48e3T^{2} \)
71 \( 1 + 88.8iT - 5.04e3T^{2} \)
73 \( 1 + 19.8T + 5.32e3T^{2} \)
79 \( 1 + 51.7iT - 6.24e3T^{2} \)
83 \( 1 + 6.62iT - 6.88e3T^{2} \)
89 \( 1 - 31.5T + 7.92e3T^{2} \)
97 \( 1 - 159.T + 9.40e3T^{2} \)
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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

−14.95317739041712193718338656280, −13.76593808018077095273042395845, −12.45665226699905645564089303446, −11.79542393712047135110573364661, −9.857346453974866751817614550592, −8.736617445591436913774944275187, −7.75205888824770045185724092610, −6.14846673263707563468663741264, −5.09630037822574433561156502421, −3.39654361255500520752466058899, 1.19273890496772193541549050874, 3.66341852632932273931561708643, 4.69463370649180075740056518270, 6.87641928759963806792994962594, 8.007770747370684216401322367292, 10.06728965408119113640426873386, 10.30246396822755154553989418433, 11.86787070347395798583556112125, 12.68922090160238752619757165578, 13.72660024535126120281232831682

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