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Properties

Label 2-312-104.77-c3-0-28
Degree $2$
Conductor $312$
Sign $-0.460 - 0.887i$
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  + (−1.04 + 2.62i)2-s + 3i·3-s + (−5.81 − 5.49i)4-s − 7.49·5-s + (−7.88 − 3.13i)6-s − 9.78i·7-s + (20.5 − 9.53i)8-s − 9·9-s + (7.83 − 19.6i)10-s + 72.0·11-s + (16.4 − 17.4i)12-s + (−46.8 + 2.05i)13-s + (25.7 + 10.2i)14-s − 22.4i·15-s + (3.59 + 63.8i)16-s + 14.0·17-s + ⋯
L(s)  = 1  + (−0.369 + 0.929i)2-s + 0.577i·3-s + (−0.726 − 0.686i)4-s − 0.670·5-s + (−0.536 − 0.213i)6-s − 0.528i·7-s + (0.906 − 0.421i)8-s − 0.333·9-s + (0.247 − 0.622i)10-s + 1.97·11-s + (0.396 − 0.419i)12-s + (−0.999 + 0.0439i)13-s + (0.490 + 0.195i)14-s − 0.386i·15-s + (0.0561 + 0.998i)16-s + 0.200·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.460 - 0.887i$
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: \((2,\ 312,\ (\ :3/2),\ -0.460 - 0.887i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.184505123\)
\(L(\frac12)\) \(\approx\) \(1.184505123\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 - 2.62i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (46.8 - 2.05i)T \)
good5 \( 1 + 7.49T + 125T^{2} \)
7 \( 1 + 9.78iT - 343T^{2} \)
11 \( 1 - 72.0T + 1.33e3T^{2} \)
17 \( 1 - 14.0T + 4.91e3T^{2} \)
19 \( 1 - 126.T + 6.85e3T^{2} \)
23 \( 1 + 74.7T + 1.21e4T^{2} \)
29 \( 1 - 177. iT - 2.43e4T^{2} \)
31 \( 1 - 204. iT - 2.97e4T^{2} \)
37 \( 1 - 262.T + 5.06e4T^{2} \)
41 \( 1 + 376. iT - 6.89e4T^{2} \)
43 \( 1 - 179. iT - 7.95e4T^{2} \)
47 \( 1 - 4.21iT - 1.03e5T^{2} \)
53 \( 1 - 102. iT - 1.48e5T^{2} \)
59 \( 1 - 231.T + 2.05e5T^{2} \)
61 \( 1 - 876. iT - 2.26e5T^{2} \)
67 \( 1 + 597.T + 3.00e5T^{2} \)
71 \( 1 - 576. iT - 3.57e5T^{2} \)
73 \( 1 - 657. iT - 3.89e5T^{2} \)
79 \( 1 + 137.T + 4.93e5T^{2} \)
83 \( 1 - 1.08e3T + 5.71e5T^{2} \)
89 \( 1 + 269. iT - 7.04e5T^{2} \)
97 \( 1 - 1.73e3iT - 9.12e5T^{2} \)
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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

−11.53947567895673986035317535469, −10.30663427965901806444364974154, −9.513674458265010595060923729163, −8.791317198646860638801246243497, −7.54989052839127265544126281062, −6.94116694288333441326695923553, −5.65585945130952517219206984440, −4.45903637342246737480387136350, −3.65635492441444926701992801445, −1.06249392947142418816384505738, 0.64972694753019423033842158365, 2.03487049054812027590155232920, 3.41703682906410060581237309132, 4.47758480108955316600700040135, 6.03996142586831235402914977926, 7.42073823198605277973640575224, 8.072907015690593536248147135230, 9.340838262681964120025918721630, 9.741027373783726730419614075139, 11.45683992653048152170014871932

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