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Properties

Label 2-76-76.75-c3-0-12
Degree $2$
Conductor $76$
Sign $0.488 + 0.872i$
Analytic cond. $4.48414$
Root an. cond. $2.11758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.603 + 2.76i)2-s − 8.93·3-s + (−7.27 + 3.33i)4-s + 6.23·5-s + (−5.39 − 24.6i)6-s − 11.1i·7-s + (−13.6 − 18.0i)8-s + 52.8·9-s + (3.76 + 17.2i)10-s − 15.9i·11-s + (64.9 − 29.8i)12-s − 34.4i·13-s + (30.6 − 6.70i)14-s − 55.7·15-s + (41.7 − 48.5i)16-s − 97.5·17-s + ⋯
L(s)  = 1  + (0.213 + 0.976i)2-s − 1.71·3-s + (−0.908 + 0.417i)4-s + 0.557·5-s + (−0.367 − 1.67i)6-s − 0.599i·7-s + (−0.601 − 0.798i)8-s + 1.95·9-s + (0.119 + 0.544i)10-s − 0.437i·11-s + (1.56 − 0.717i)12-s − 0.734i·13-s + (0.585 − 0.127i)14-s − 0.958·15-s + (0.652 − 0.758i)16-s − 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ 0.488 + 0.872i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.395235 - 0.231572i\)
\(L(\frac12)\) \(\approx\) \(0.395235 - 0.231572i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.603 - 2.76i)T \)
19 \( 1 + (6.66 + 82.5i)T \)
good3 \( 1 + 8.93T + 27T^{2} \)
5 \( 1 - 6.23T + 125T^{2} \)
7 \( 1 + 11.1iT - 343T^{2} \)
11 \( 1 + 15.9iT - 1.33e3T^{2} \)
13 \( 1 + 34.4iT - 2.19e3T^{2} \)
17 \( 1 + 97.5T + 4.91e3T^{2} \)
23 \( 1 - 96.9iT - 1.21e4T^{2} \)
29 \( 1 + 222. iT - 2.43e4T^{2} \)
31 \( 1 + 91.0T + 2.97e4T^{2} \)
37 \( 1 + 395. iT - 5.06e4T^{2} \)
41 \( 1 - 266. iT - 6.89e4T^{2} \)
43 \( 1 - 384. iT - 7.95e4T^{2} \)
47 \( 1 + 400. iT - 1.03e5T^{2} \)
53 \( 1 - 201. iT - 1.48e5T^{2} \)
59 \( 1 + 18.7T + 2.05e5T^{2} \)
61 \( 1 + 400.T + 2.26e5T^{2} \)
67 \( 1 + 839.T + 3.00e5T^{2} \)
71 \( 1 - 855.T + 3.57e5T^{2} \)
73 \( 1 + 754.T + 3.89e5T^{2} \)
79 \( 1 - 533.T + 4.93e5T^{2} \)
83 \( 1 + 1.09e3iT - 5.71e5T^{2} \)
89 \( 1 - 129. iT - 7.04e5T^{2} \)
97 \( 1 - 681. iT - 9.12e5T^{2} \)
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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

−13.55129080423995333143757630374, −13.05174443063306945889350110209, −11.61551901810540037399323462470, −10.60266737873186530520316473113, −9.353413362306176803041467848312, −7.52760217089400309406840050193, −6.38143741590710333896603101796, −5.57288561627090188985498723897, −4.35181761266017310987411900707, −0.33624211939320344980608043802, 1.82116473816113658589766040196, 4.42804914915384075389572955571, 5.53443652835112373046639403018, 6.58830397225067249689192109889, 8.975187884561336799607189490695, 10.17836098313354258973821571130, 11.00189120959966313673710736908, 12.01760282005435591465496864610, 12.60917252016268164013462243142, 13.83246537094042365804972193882

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