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Properties

Label 2-312-104.77-c3-0-35
Degree $2$
Conductor $312$
Sign $-0.639 - 0.768i$
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  + (2.56 + 1.18i)2-s + 3i·3-s + (5.20 + 6.07i)4-s + 8.43·5-s + (−3.54 + 7.70i)6-s + 8.36i·7-s + (6.20 + 21.7i)8-s − 9·9-s + (21.6 + 9.97i)10-s − 57.7·11-s + (−18.2 + 15.6i)12-s + (18.9 + 42.8i)13-s + (−9.88 + 21.5i)14-s + 25.3i·15-s + (−9.78 + 63.2i)16-s − 72.1·17-s + ⋯
L(s)  = 1  + (0.908 + 0.417i)2-s + 0.577i·3-s + (0.650 + 0.759i)4-s + 0.754·5-s + (−0.241 + 0.524i)6-s + 0.451i·7-s + (0.274 + 0.961i)8-s − 0.333·9-s + (0.685 + 0.315i)10-s − 1.58·11-s + (−0.438 + 0.375i)12-s + (0.404 + 0.914i)13-s + (−0.188 + 0.410i)14-s + 0.435i·15-s + (−0.152 + 0.988i)16-s − 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.639 - 0.768i$
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: \((2,\ 312,\ (\ :3/2),\ -0.639 - 0.768i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.248111219\)
\(L(\frac12)\) \(\approx\) \(3.248111219\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.56 - 1.18i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (-18.9 - 42.8i)T \)
good5 \( 1 - 8.43T + 125T^{2} \)
7 \( 1 - 8.36iT - 343T^{2} \)
11 \( 1 + 57.7T + 1.33e3T^{2} \)
17 \( 1 + 72.1T + 4.91e3T^{2} \)
19 \( 1 - 144.T + 6.85e3T^{2} \)
23 \( 1 - 168.T + 1.21e4T^{2} \)
29 \( 1 + 96.0iT - 2.43e4T^{2} \)
31 \( 1 - 59.3iT - 2.97e4T^{2} \)
37 \( 1 + 187.T + 5.06e4T^{2} \)
41 \( 1 - 211. iT - 6.89e4T^{2} \)
43 \( 1 + 160. iT - 7.95e4T^{2} \)
47 \( 1 - 539. iT - 1.03e5T^{2} \)
53 \( 1 + 583. iT - 1.48e5T^{2} \)
59 \( 1 + 236.T + 2.05e5T^{2} \)
61 \( 1 + 438. iT - 2.26e5T^{2} \)
67 \( 1 - 639.T + 3.00e5T^{2} \)
71 \( 1 - 79.3iT - 3.57e5T^{2} \)
73 \( 1 + 40.6iT - 3.89e5T^{2} \)
79 \( 1 - 807.T + 4.93e5T^{2} \)
83 \( 1 - 1.39e3T + 5.71e5T^{2} \)
89 \( 1 + 1.05e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.00e3iT - 9.12e5T^{2} \)
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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

−11.53666799650851135170409518316, −10.87584106206286645340696038962, −9.682589747089382430201442678124, −8.753562279185520140106136995720, −7.59261388393308480572340229970, −6.44712784942574454934731994346, −5.39921920182866777775900762440, −4.81286722136832984503517846636, −3.27692809030279181221359495473, −2.20858176488484170612119259192, 0.865551365594580357964064637576, 2.32694769031476200239141208591, 3.32471884527527495618411211583, 5.09917030626934351509719817184, 5.61956966384707531022248071384, 6.90032764023071160689080763960, 7.71951610740931399701460647240, 9.199092010495217900981986228711, 10.40621136120507637093999510939, 10.84159398357218634775754806856

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