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Properties

Label 2-76-76.75-c1-0-5
Degree $2$
Conductor $76$
Sign $0.603 + 0.797i$
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  + (1.06 − 0.927i)2-s − 1.19·3-s + (0.280 − 1.98i)4-s + 1.56·5-s + (−1.28 + 1.11i)6-s + 0.868i·7-s + (−1.53 − 2.37i)8-s − 1.56·9-s + (1.66 − 1.44i)10-s + 3.09i·11-s + (−0.336 + 2.37i)12-s + 4.74i·13-s + (0.804 + 0.927i)14-s − 1.87·15-s + (−3.84 − 1.11i)16-s − 17-s + ⋯
L(s)  = 1  + (0.755 − 0.655i)2-s − 0.692·3-s + (0.140 − 0.990i)4-s + 0.698·5-s + (−0.522 + 0.453i)6-s + 0.328i·7-s + (−0.543 − 0.839i)8-s − 0.520·9-s + (0.527 − 0.457i)10-s + 0.932i·11-s + (−0.0972 + 0.685i)12-s + 1.31i·13-s + (0.215 + 0.247i)14-s − 0.483·15-s + (−0.960 − 0.277i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.603 + 0.797i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03052 - 0.512508i\)
\(L(\frac12)\) \(\approx\) \(1.03052 - 0.512508i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 + 0.927i)T \)
19 \( 1 + (-3.07 + 3.09i)T \)
good3 \( 1 + 1.19T + 3T^{2} \)
5 \( 1 - 1.56T + 5T^{2} \)
7 \( 1 - 0.868iT - 7T^{2} \)
11 \( 1 - 3.09iT - 11T^{2} \)
13 \( 1 - 4.74iT - 13T^{2} \)
17 \( 1 + T + 17T^{2} \)
23 \( 1 + 3.96iT - 23T^{2} \)
29 \( 1 + 8.45iT - 29T^{2} \)
31 \( 1 + 4.27T + 31T^{2} \)
37 \( 1 - 3.70iT - 37T^{2} \)
41 \( 1 + 3.70iT - 41T^{2} \)
43 \( 1 - 11.0iT - 43T^{2} \)
47 \( 1 + 9.27iT - 47T^{2} \)
53 \( 1 - 1.04iT - 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 + 0.684T + 61T^{2} \)
67 \( 1 + 9.74T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 8.12T + 73T^{2} \)
79 \( 1 + 8.01T + 79T^{2} \)
83 \( 1 - 9.65iT - 83T^{2} \)
89 \( 1 + 5.79iT - 89T^{2} \)
97 \( 1 - 16.9iT - 97T^{2} \)
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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

−14.15056735641180988280449719782, −13.24916265150195017301943112562, −11.98793472986176145255184472531, −11.42202548566248900053695846140, −10.09689573793728439140077686579, −9.132640273946707744970785067147, −6.76656881913769197688988453338, −5.71508598516158499356213175893, −4.50781930093701196262537216478, −2.27673060046378987766924326058, 3.32472764545595268802635285011, 5.42207377499383622179197897989, 5.88743290187322996826442695187, 7.44800365659175530128906231750, 8.784772504140371997895987787647, 10.50749616570065624228243596259, 11.54757108419020966380362268511, 12.72411917044511659573645710998, 13.69803480950288826711553840107, 14.49195505430138544548757172363

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