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Properties

Label 2-312-104.77-c3-0-36
Degree $2$
Conductor $312$
Sign $0.580 + 0.814i$
Analytic cond. $18.4085$
Root an. cond. $4.29052$
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  + (−1.62 + 2.31i)2-s − 3i·3-s + (−2.71 − 7.52i)4-s − 20.0·5-s + (6.94 + 4.87i)6-s + 33.1i·7-s + (21.8 + 5.93i)8-s − 9·9-s + (32.5 − 46.4i)10-s + 7.50·11-s + (−22.5 + 8.15i)12-s + (−29.7 + 36.2i)13-s + (−76.7 − 53.8i)14-s + 60.1i·15-s + (−49.2 + 40.9i)16-s + 88.2·17-s + ⋯
L(s)  = 1  + (−0.574 + 0.818i)2-s − 0.577i·3-s + (−0.339 − 0.940i)4-s − 1.79·5-s + (0.472 + 0.331i)6-s + 1.78i·7-s + (0.964 + 0.262i)8-s − 0.333·9-s + (1.03 − 1.46i)10-s + 0.205·11-s + (−0.543 + 0.196i)12-s + (−0.633 + 0.773i)13-s + (−1.46 − 1.02i)14-s + 1.03i·15-s + (−0.769 + 0.639i)16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.580 + 0.814i$
Analytic conductor: \(18.4085\)
Root analytic conductor: \(4.29052\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :3/2),\ 0.580 + 0.814i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3061681829\)
\(L(\frac12)\) \(\approx\) \(0.3061681829\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.62 - 2.31i)T \)
3 \( 1 + 3iT \)
13 \( 1 + (29.7 - 36.2i)T \)
good5 \( 1 + 20.0T + 125T^{2} \)
7 \( 1 - 33.1iT - 343T^{2} \)
11 \( 1 - 7.50T + 1.33e3T^{2} \)
17 \( 1 - 88.2T + 4.91e3T^{2} \)
19 \( 1 + 43.0T + 6.85e3T^{2} \)
23 \( 1 + 192.T + 1.21e4T^{2} \)
29 \( 1 + 133. iT - 2.43e4T^{2} \)
31 \( 1 + 114. iT - 2.97e4T^{2} \)
37 \( 1 + 59.6T + 5.06e4T^{2} \)
41 \( 1 + 505. iT - 6.89e4T^{2} \)
43 \( 1 - 142. iT - 7.95e4T^{2} \)
47 \( 1 - 97.2iT - 1.03e5T^{2} \)
53 \( 1 + 141. iT - 1.48e5T^{2} \)
59 \( 1 - 794.T + 2.05e5T^{2} \)
61 \( 1 + 124. iT - 2.26e5T^{2} \)
67 \( 1 - 174.T + 3.00e5T^{2} \)
71 \( 1 + 519. iT - 3.57e5T^{2} \)
73 \( 1 - 616. iT - 3.89e5T^{2} \)
79 \( 1 - 937.T + 4.93e5T^{2} \)
83 \( 1 + 1.14e3T + 5.71e5T^{2} \)
89 \( 1 + 15.8iT - 7.04e5T^{2} \)
97 \( 1 - 481. iT - 9.12e5T^{2} \)
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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

−11.41955932198992060415819452813, −9.885852090845465862997375413131, −8.810091560858840855663683832173, −8.145085704381023186366153719456, −7.48640637171887984391151094749, −6.35631574649420557245787892110, −5.35635971193052173852542274783, −4.00869392754295006160156270524, −2.17797659682752133883260165592, −0.19377890161410727837721888264, 0.871673197118746359271607380181, 3.31306934738544358187255432533, 3.88449788971339079094683441742, 4.73063234444842749042329282155, 7.08157595550907626811304158607, 7.80569403461171258238389131636, 8.376483940315759307022547620621, 9.964253457650350038394627941844, 10.39206071674242216222629301601, 11.25031747195643483468856702493

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