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Properties

Label 2-76-76.7-c2-0-0
Degree $2$
Conductor $76$
Sign $0.216 - 0.976i$
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  + (−1.20 − 1.59i)2-s + (−1.67 + 0.967i)3-s + (−1.11 + 3.84i)4-s + (−1.80 − 3.12i)5-s + (3.55 + 1.51i)6-s + 13.2i·7-s + (7.48 − 2.82i)8-s + (−2.62 + 4.55i)9-s + (−2.83 + 6.64i)10-s + 2.62i·11-s + (−1.84 − 7.51i)12-s + (−0.424 + 0.735i)13-s + (21.1 − 15.8i)14-s + (6.04 + 3.49i)15-s + (−13.5 − 8.57i)16-s + (6.24 + 10.8i)17-s + ⋯
L(s)  = 1  + (−0.600 − 0.799i)2-s + (−0.558 + 0.322i)3-s + (−0.279 + 0.960i)4-s + (−0.361 − 0.625i)5-s + (0.593 + 0.253i)6-s + 1.89i·7-s + (0.935 − 0.353i)8-s + (−0.292 + 0.505i)9-s + (−0.283 + 0.664i)10-s + 0.239i·11-s + (−0.153 − 0.626i)12-s + (−0.0326 + 0.0565i)13-s + (1.51 − 1.13i)14-s + (0.403 + 0.232i)15-s + (−0.844 − 0.536i)16-s + (0.367 + 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.216 - 0.976i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.216 - 0.976i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.395276 + 0.317344i\)
\(L(\frac12)\) \(\approx\) \(0.395276 + 0.317344i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 + 1.59i)T \)
19 \( 1 + (18.2 - 5.21i)T \)
good3 \( 1 + (1.67 - 0.967i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.80 + 3.12i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 13.2iT - 49T^{2} \)
11 \( 1 - 2.62iT - 121T^{2} \)
13 \( 1 + (0.424 - 0.735i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-6.24 - 10.8i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (26.9 + 15.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (9.98 - 17.2i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 - 28.0iT - 961T^{2} \)
37 \( 1 - 61.9T + 1.36e3T^{2} \)
41 \( 1 + (-10.4 - 18.1i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-23.9 + 13.8i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-44.2 - 25.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-9.57 + 16.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-63.8 + 36.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (32.5 - 56.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-21.6 - 12.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (32.3 - 18.6i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-15.9 - 27.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (82.1 - 47.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 22.6iT - 6.88e3T^{2} \)
89 \( 1 + (-60.9 + 105. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (10.1 + 17.6i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.57752963673363513412592121241, −12.70670585951209076986736633181, −12.24671966740012346133686667504, −11.28161490534198022109980657175, −10.11801229469531354819287412340, −8.798611065514666504094971027166, −8.173995578074701930208866727635, −5.88398001737506342431312286202, −4.49551161112876136164921994684, −2.34628247189673899608430213338, 0.55972999119163929565116119162, 4.08415045922211444870892901588, 6.00330827679970372667366907336, 7.07571995885826852341617705830, 7.80603752515716489321882561932, 9.566649586417859860223804771320, 10.68631626493073523416971667319, 11.44117191587477414086277096730, 13.26168468037497222705251706834, 14.19401007270897689453764818782

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