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Properties

Label 2-76-4.3-c2-0-2
Degree $2$
Conductor $76$
Sign $0.529 - 0.848i$
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  + (−1.92 − 0.551i)2-s + 0.644i·3-s + (3.39 + 2.11i)4-s − 2.32·5-s + (0.355 − 1.23i)6-s + 8.62i·7-s + (−5.35 − 5.94i)8-s + 8.58·9-s + (4.47 + 1.28i)10-s + 19.2i·11-s + (−1.36 + 2.18i)12-s + 13.8·13-s + (4.75 − 16.5i)14-s − 1.50i·15-s + (7.01 + 14.3i)16-s − 12.5·17-s + ⋯
L(s)  = 1  + (−0.961 − 0.275i)2-s + 0.214i·3-s + (0.848 + 0.529i)4-s − 0.465·5-s + (0.0592 − 0.206i)6-s + 1.23i·7-s + (−0.669 − 0.743i)8-s + 0.953·9-s + (0.447 + 0.128i)10-s + 1.75i·11-s + (−0.113 + 0.182i)12-s + 1.06·13-s + (0.339 − 1.18i)14-s − 0.100i·15-s + (0.438 + 0.898i)16-s − 0.736·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.529 - 0.848i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.529 - 0.848i$
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: \((2,\ 76,\ (\ :1),\ 0.529 - 0.848i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.685531 + 0.380019i\)
\(L(\frac12)\) \(\approx\) \(0.685531 + 0.380019i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.92 + 0.551i)T \)
19 \( 1 - 4.35iT \)
good3 \( 1 - 0.644iT - 9T^{2} \)
5 \( 1 + 2.32T + 25T^{2} \)
7 \( 1 - 8.62iT - 49T^{2} \)
11 \( 1 - 19.2iT - 121T^{2} \)
13 \( 1 - 13.8T + 169T^{2} \)
17 \( 1 + 12.5T + 289T^{2} \)
23 \( 1 + 37.4iT - 529T^{2} \)
29 \( 1 + 6.36T + 841T^{2} \)
31 \( 1 + 5.44iT - 961T^{2} \)
37 \( 1 - 20.9T + 1.36e3T^{2} \)
41 \( 1 - 72.8T + 1.68e3T^{2} \)
43 \( 1 - 10.1iT - 1.84e3T^{2} \)
47 \( 1 + 32.4iT - 2.20e3T^{2} \)
53 \( 1 + 42.9T + 2.80e3T^{2} \)
59 \( 1 - 38.9iT - 3.48e3T^{2} \)
61 \( 1 - 25.5T + 3.72e3T^{2} \)
67 \( 1 + 65.3iT - 4.48e3T^{2} \)
71 \( 1 + 18.7iT - 5.04e3T^{2} \)
73 \( 1 - 72.8T + 5.32e3T^{2} \)
79 \( 1 + 139. iT - 6.24e3T^{2} \)
83 \( 1 - 94.7iT - 6.88e3T^{2} \)
89 \( 1 - 33.3T + 7.92e3T^{2} \)
97 \( 1 + 150.T + 9.40e3T^{2} \)
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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

−15.02833996743646916036593290485, −12.81896318262718657265444348277, −12.20803491498449282541504608929, −11.00246897889489753904736309306, −9.840579256050574539898149544069, −8.912174493969545085759749211738, −7.69525487834786033340873213967, −6.41286588337336248692465832592, −4.26075125748547407591803018228, −2.13426657726474913408405289393, 0.994023242608079120389770047462, 3.77218193780077853813295469411, 6.06345320599603749332059786030, 7.29099521408902766318861371250, 8.187942712108214051665373525606, 9.499272513154555161084873354197, 10.87892193705116623491288757846, 11.32880480924131976098159479007, 13.21038179852798647652662591746, 14.01290697398621490682605103148

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