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Properties

Label 2-76-4.3-c2-0-6
Degree $2$
Conductor $76$
Sign $0.866 + 0.5i$
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 − 1.73i)2-s + 1.31i·3-s + (−1.99 + 3.46i)4-s + 6.54·5-s + (2.27 − 1.31i)6-s − 1.31i·7-s + 7.99·8-s + 7.27·9-s + (−6.54 − 11.3i)10-s − 4.30i·11-s + (−4.54 − 2.62i)12-s + 0.824·13-s + (−2.27 + 1.31i)14-s + 8.60i·15-s + (−8 − 13.8i)16-s + 0.274·17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 0.437i·3-s + (−0.499 + 0.866i)4-s + 1.30·5-s + (0.379 − 0.218i)6-s − 0.187i·7-s + 0.999·8-s + 0.808·9-s + (−0.654 − 1.13i)10-s − 0.391i·11-s + (−0.379 − 0.218i)12-s + 0.0634·13-s + (−0.162 + 0.0938i)14-s + 0.573i·15-s + (−0.5 − 0.866i)16-s + 0.0161·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.866 + 0.5i$
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: \((2,\ 76,\ (\ :1),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.13185 - 0.303279i\)
\(L(\frac12)\) \(\approx\) \(1.13185 - 0.303279i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
19 \( 1 - 4.35iT \)
good3 \( 1 - 1.31iT - 9T^{2} \)
5 \( 1 - 6.54T + 25T^{2} \)
7 \( 1 + 1.31iT - 49T^{2} \)
11 \( 1 + 4.30iT - 121T^{2} \)
13 \( 1 - 0.824T + 169T^{2} \)
17 \( 1 - 0.274T + 289T^{2} \)
23 \( 1 - 33.5iT - 529T^{2} \)
29 \( 1 + 33.3T + 841T^{2} \)
31 \( 1 + 48.7iT - 961T^{2} \)
37 \( 1 + 36.1T + 1.36e3T^{2} \)
41 \( 1 + 68.7T + 1.68e3T^{2} \)
43 \( 1 + 55.6iT - 1.84e3T^{2} \)
47 \( 1 + 24.1iT - 2.20e3T^{2} \)
53 \( 1 - 25.9T + 2.80e3T^{2} \)
59 \( 1 - 46.2iT - 3.48e3T^{2} \)
61 \( 1 + 47.0T + 3.72e3T^{2} \)
67 \( 1 - 112. iT - 4.48e3T^{2} \)
71 \( 1 + 70.4iT - 5.04e3T^{2} \)
73 \( 1 + 58.0T + 5.32e3T^{2} \)
79 \( 1 + 113. iT - 6.24e3T^{2} \)
83 \( 1 - 148. iT - 6.88e3T^{2} \)
89 \( 1 - 57.8T + 7.92e3T^{2} \)
97 \( 1 + 117.T + 9.40e3T^{2} \)
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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

−13.64509750007543749174499083976, −13.26955133985365298640384988987, −11.81932286017845616320378244121, −10.53204233362395266473197041057, −9.842156567191132062974688842440, −8.975746652778786052636048396828, −7.33163752798424649704836560897, −5.48063168836927258090270426209, −3.75955991938948054450110569198, −1.79739967030431962405015033792, 1.73985840038128703296227782032, 4.91431741519517579633433478371, 6.25004596022357815922488691365, 7.15091971872873401847872504131, 8.647610547000475939808125906103, 9.738576406944993851754355580447, 10.55490795664930613190493114250, 12.54648027515165091758505538752, 13.47329082533679930266515411046, 14.35485709603682708150432961548

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