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Properties

Label 2-312-104.77-c3-0-45
Degree $2$
Conductor $312$
Sign $0.981 - 0.191i$
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  + (0.800 + 2.71i)2-s + 3i·3-s + (−6.71 + 4.34i)4-s − 12.1·5-s + (−8.13 + 2.40i)6-s − 21.1i·7-s + (−17.1 − 14.7i)8-s − 9·9-s + (−9.70 − 32.8i)10-s − 16.0·11-s + (−13.0 − 20.1i)12-s + (36.8 + 29.0i)13-s + (57.3 − 16.9i)14-s − 36.3i·15-s + (26.2 − 58.3i)16-s + 45.8·17-s + ⋯
L(s)  = 1  + (0.283 + 0.959i)2-s + 0.577i·3-s + (−0.839 + 0.543i)4-s − 1.08·5-s + (−0.553 + 0.163i)6-s − 1.14i·7-s + (−0.758 − 0.651i)8-s − 0.333·9-s + (−0.307 − 1.04i)10-s − 0.438·11-s + (−0.313 − 0.484i)12-s + (0.785 + 0.619i)13-s + (1.09 − 0.323i)14-s − 0.626i·15-s + (0.410 − 0.911i)16-s + 0.653·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.981 - 0.191i$
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: \((2,\ 312,\ (\ :3/2),\ 0.981 - 0.191i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.108213021\)
\(L(\frac12)\) \(\approx\) \(1.108213021\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.800 - 2.71i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (-36.8 - 29.0i)T \)
good5 \( 1 + 12.1T + 125T^{2} \)
7 \( 1 + 21.1iT - 343T^{2} \)
11 \( 1 + 16.0T + 1.33e3T^{2} \)
17 \( 1 - 45.8T + 4.91e3T^{2} \)
19 \( 1 - 117.T + 6.85e3T^{2} \)
23 \( 1 + 97.3T + 1.21e4T^{2} \)
29 \( 1 + 90.0iT - 2.43e4T^{2} \)
31 \( 1 + 151. iT - 2.97e4T^{2} \)
37 \( 1 - 66.3T + 5.06e4T^{2} \)
41 \( 1 + 112. iT - 6.89e4T^{2} \)
43 \( 1 - 234. iT - 7.95e4T^{2} \)
47 \( 1 + 580. iT - 1.03e5T^{2} \)
53 \( 1 + 241. iT - 1.48e5T^{2} \)
59 \( 1 + 313.T + 2.05e5T^{2} \)
61 \( 1 + 334. iT - 2.26e5T^{2} \)
67 \( 1 - 315.T + 3.00e5T^{2} \)
71 \( 1 + 503. iT - 3.57e5T^{2} \)
73 \( 1 + 757. iT - 3.89e5T^{2} \)
79 \( 1 - 1.13e3T + 4.93e5T^{2} \)
83 \( 1 - 822.T + 5.71e5T^{2} \)
89 \( 1 - 803. iT - 7.04e5T^{2} \)
97 \( 1 + 750. iT - 9.12e5T^{2} \)
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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

−11.38726237930981107812993573050, −10.21080278975245015950399540488, −9.308019887405266388811556703074, −7.958153594103724398073692045993, −7.66637877493551350487542485494, −6.42472824895634000112657444436, −5.16614532847081868814717754864, −4.03281499983128230546853283800, −3.54828017135141183636477276196, −0.45549523602458334076390992220, 1.12670991228808916640224381920, 2.73514982208916658908314234438, 3.63486254602505085011552770787, 5.16315605146332413448515485369, 5.97846728247275241569707987319, 7.67109629916802693813080753099, 8.371983321286301091237631080404, 9.359461778270892024458268776846, 10.56096707133710763364110041882, 11.50814569656800093319868071667

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