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Properties

Label 2-312-104.77-c3-0-25
Degree $2$
Conductor $312$
Sign $-0.996 - 0.0860i$
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  + (2.48 + 1.34i)2-s + 3i·3-s + (4.36 + 6.70i)4-s − 8.37·5-s + (−4.04 + 7.46i)6-s + 16.5i·7-s + (1.83 + 22.5i)8-s − 9·9-s + (−20.8 − 11.2i)10-s + 23.2·11-s + (−20.1 + 13.1i)12-s + (46.2 + 7.80i)13-s + (−22.3 + 41.2i)14-s − 25.1i·15-s + (−25.8 + 58.5i)16-s − 14.7·17-s + ⋯
L(s)  = 1  + (0.879 + 0.476i)2-s + 0.577i·3-s + (0.546 + 0.837i)4-s − 0.748·5-s + (−0.275 + 0.507i)6-s + 0.896i·7-s + (0.0811 + 0.996i)8-s − 0.333·9-s + (−0.658 − 0.356i)10-s + 0.636·11-s + (−0.483 + 0.315i)12-s + (0.986 + 0.166i)13-s + (−0.426 + 0.787i)14-s − 0.432i·15-s + (−0.403 + 0.914i)16-s − 0.210·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0860i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.996 - 0.0860i$
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: \((2,\ 312,\ (\ :3/2),\ -0.996 - 0.0860i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.163408246\)
\(L(\frac12)\) \(\approx\) \(2.163408246\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.48 - 1.34i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (-46.2 - 7.80i)T \)
good5 \( 1 + 8.37T + 125T^{2} \)
7 \( 1 - 16.5iT - 343T^{2} \)
11 \( 1 - 23.2T + 1.33e3T^{2} \)
17 \( 1 + 14.7T + 4.91e3T^{2} \)
19 \( 1 + 124.T + 6.85e3T^{2} \)
23 \( 1 + 173.T + 1.21e4T^{2} \)
29 \( 1 + 126. iT - 2.43e4T^{2} \)
31 \( 1 + 27.9iT - 2.97e4T^{2} \)
37 \( 1 - 362.T + 5.06e4T^{2} \)
41 \( 1 - 309. iT - 6.89e4T^{2} \)
43 \( 1 - 556. iT - 7.95e4T^{2} \)
47 \( 1 + 163. iT - 1.03e5T^{2} \)
53 \( 1 + 334. iT - 1.48e5T^{2} \)
59 \( 1 - 809.T + 2.05e5T^{2} \)
61 \( 1 + 6.20iT - 2.26e5T^{2} \)
67 \( 1 - 252.T + 3.00e5T^{2} \)
71 \( 1 - 822. iT - 3.57e5T^{2} \)
73 \( 1 - 449. iT - 3.89e5T^{2} \)
79 \( 1 + 547.T + 4.93e5T^{2} \)
83 \( 1 - 959.T + 5.71e5T^{2} \)
89 \( 1 - 656. iT - 7.04e5T^{2} \)
97 \( 1 - 417. iT - 9.12e5T^{2} \)
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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

−11.52879874462655633069649771878, −11.34681939070531383705730210832, −9.788313616366043376070553090631, −8.511456514363158702171059401776, −8.054899234184950866309436433126, −6.44569484078351549174391106035, −5.87303108264135567945577077580, −4.37010615724869702824724532413, −3.84714088051075071488881255599, −2.34929645918073698974987757608, 0.57432837841630311973721317642, 1.98616956370822765938944813801, 3.72361737186202038213536876384, 4.21248545600705779207786940540, 5.88109479339875727902185173974, 6.71480831637061740037282764632, 7.69816710341010240444017087584, 8.828554111890545078075309473181, 10.31249880282825864895902685794, 10.97755155254204262398737829132

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