Properties

Label 2-76-19.14-c2-0-0
Degree $2$
Conductor $76$
Sign $0.533 - 0.845i$
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  + (−4.45 − 0.786i)3-s + (5.56 + 4.66i)5-s + (4.24 + 7.35i)7-s + (10.8 + 3.93i)9-s + (−4.58 + 7.94i)11-s + (8.90 − 1.57i)13-s + (−21.1 − 25.1i)15-s + (−6.76 + 2.46i)17-s + (−18.6 + 3.58i)19-s + (−13.1 − 36.1i)21-s + (28.1 − 23.6i)23-s + (4.81 + 27.2i)25-s + (−9.83 − 5.68i)27-s + (−16.9 + 46.6i)29-s + (30.9 − 17.8i)31-s + ⋯
L(s)  = 1  + (−1.48 − 0.262i)3-s + (1.11 + 0.933i)5-s + (0.606 + 1.05i)7-s + (1.20 + 0.437i)9-s + (−0.416 + 0.722i)11-s + (0.685 − 0.120i)13-s + (−1.40 − 1.67i)15-s + (−0.398 + 0.144i)17-s + (−0.982 + 0.188i)19-s + (−0.626 − 1.72i)21-s + (1.22 − 1.02i)23-s + (0.192 + 1.09i)25-s + (−0.364 − 0.210i)27-s + (−0.585 + 1.60i)29-s + (0.998 − 0.576i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.533 - 0.845i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.825906 + 0.455586i\)
\(L(\frac12)\) \(\approx\) \(0.825906 + 0.455586i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (18.6 - 3.58i)T \)
good3 \( 1 + (4.45 + 0.786i)T + (8.45 + 3.07i)T^{2} \)
5 \( 1 + (-5.56 - 4.66i)T + (4.34 + 24.6i)T^{2} \)
7 \( 1 + (-4.24 - 7.35i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (4.58 - 7.94i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-8.90 + 1.57i)T + (158. - 57.8i)T^{2} \)
17 \( 1 + (6.76 - 2.46i)T + (221. - 185. i)T^{2} \)
23 \( 1 + (-28.1 + 23.6i)T + (91.8 - 520. i)T^{2} \)
29 \( 1 + (16.9 - 46.6i)T + (-644. - 540. i)T^{2} \)
31 \( 1 + (-30.9 + 17.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 61.6iT - 1.36e3T^{2} \)
41 \( 1 + (21.8 + 3.84i)T + (1.57e3 + 574. i)T^{2} \)
43 \( 1 + (50.2 + 42.1i)T + (321. + 1.82e3i)T^{2} \)
47 \( 1 + (-67.7 - 24.6i)T + (1.69e3 + 1.41e3i)T^{2} \)
53 \( 1 + (-5.17 - 6.17i)T + (-487. + 2.76e3i)T^{2} \)
59 \( 1 + (-3.10 - 8.52i)T + (-2.66e3 + 2.23e3i)T^{2} \)
61 \( 1 + (-32.9 + 27.6i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (-2.86 + 7.87i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (-5.02 + 5.98i)T + (-875. - 4.96e3i)T^{2} \)
73 \( 1 + (-17.9 + 102. i)T + (-5.00e3 - 1.82e3i)T^{2} \)
79 \( 1 + (5.92 + 1.04i)T + (5.86e3 + 2.13e3i)T^{2} \)
83 \( 1 + (-17.5 - 30.3i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-56.6 + 9.98i)T + (7.44e3 - 2.70e3i)T^{2} \)
97 \( 1 + (36.6 + 100. i)T + (-7.20e3 + 6.04e3i)T^{2} \)
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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

−14.55208704382947821825962771377, −13.09136002434296922383027603453, −12.22530770382460453999384934835, −10.93725837506199745617427002525, −10.50079124001112527871940885850, −8.861706376889644080306487480685, −6.91966342464125029280309835139, −6.02438257804819083007278602180, −5.07669734740961281984975595770, −2.14977654994249382426844519535, 1.06612530648745692039017289826, 4.52820432842057797772687280617, 5.49304850298228743609249697831, 6.57874233990168421133384445747, 8.440190803942820821779617186523, 9.903627366782263099966356396514, 10.85863288548680798405885363703, 11.64308723578401219549339121477, 13.24407849921751030241274869891, 13.57802066310895441059170803437

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