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Properties

Label 2-76-76.27-c1-0-5
Degree $2$
Conductor $76$
Sign $0.944 - 0.327i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.327 + 1.37i)2-s + (1.42 − 2.47i)3-s + (−1.78 + 0.902i)4-s + (−0.139 + 0.242i)5-s + (3.86 + 1.15i)6-s + 1.55i·7-s + (−1.82 − 2.15i)8-s + (−2.57 − 4.45i)9-s + (−0.379 − 0.113i)10-s + 2.44i·11-s + (−0.317 + 5.69i)12-s + (−5.47 + 3.16i)13-s + (−2.13 + 0.509i)14-s + (0.399 + 0.691i)15-s + (2.37 − 3.22i)16-s + (2.18 − 3.78i)17-s + ⋯
L(s)  = 1  + (0.231 + 0.972i)2-s + (0.823 − 1.42i)3-s + (−0.892 + 0.451i)4-s + (−0.0625 + 0.108i)5-s + (1.57 + 0.470i)6-s + 0.586i·7-s + (−0.645 − 0.763i)8-s + (−0.857 − 1.48i)9-s + (−0.119 − 0.0357i)10-s + 0.735i·11-s + (−0.0916 + 1.64i)12-s + (−1.51 + 0.876i)13-s + (−0.570 + 0.136i)14-s + (0.103 + 0.178i)15-s + (0.593 − 0.805i)16-s + (0.529 − 0.917i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.944 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11969 + 0.188367i\)
\(L(\frac12)\) \(\approx\) \(1.11969 + 0.188367i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.327 - 1.37i)T \)
19 \( 1 + (3.17 + 2.99i)T \)
good3 \( 1 + (-1.42 + 2.47i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.139 - 0.242i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.55iT - 7T^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 + (5.47 - 3.16i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.18 + 3.78i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.71 + 3.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.695 + 0.401i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.42T + 31T^{2} \)
37 \( 1 - 4.74iT - 37T^{2} \)
41 \( 1 + (5.84 + 3.37i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.146 - 0.0845i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.19 + 0.691i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.25 + 2.45i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.05 + 1.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.58 + 2.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.00 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.74 - 3.02i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.43 - 9.41i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.00 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.99iT - 83T^{2} \)
89 \( 1 + (-7.76 + 4.48i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.77 + 1.02i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.65820264813378889729668754019, −13.62288683223872209893459646074, −12.65097668727785047655869118445, −11.97643968720825827874849009999, −9.477094371316353317525878463644, −8.598732637951076112049110880117, −7.22449682409080177905038991907, −6.88156930986466182722791626002, −4.96763409663635650445370529313, −2.64016570664415265477930387265, 2.97600718015399579114084827220, 4.10921366260136098055817218984, 5.32790007631649568049406409426, 8.065853514864967524085005723884, 9.141694713483646035651364796910, 10.27871600839276948970988226463, 10.65722846647068491855529803316, 12.28116141299503303133318877826, 13.45610383497322130485723152045, 14.63649094529354955445997723995

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