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Properties

Label 2-312-104.77-c3-0-13
Degree $2$
Conductor $312$
Sign $0.925 + 0.379i$
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  + (−0.481 − 2.78i)2-s − 3i·3-s + (−7.53 + 2.68i)4-s − 21.7·5-s + (−8.36 + 1.44i)6-s − 13.0i·7-s + (11.1 + 19.7i)8-s − 9·9-s + (10.4 + 60.5i)10-s − 21.2·11-s + (8.05 + 22.6i)12-s + (29.0 + 36.7i)13-s + (−36.3 + 6.28i)14-s + 65.1i·15-s + (49.5 − 40.4i)16-s − 41.3·17-s + ⋯
L(s)  = 1  + (−0.170 − 0.985i)2-s − 0.577i·3-s + (−0.942 + 0.335i)4-s − 1.94·5-s + (−0.568 + 0.0983i)6-s − 0.704i·7-s + (0.491 + 0.871i)8-s − 0.333·9-s + (0.330 + 1.91i)10-s − 0.583·11-s + (0.193 + 0.543i)12-s + (0.619 + 0.784i)13-s + (−0.694 + 0.120i)14-s + 1.12i·15-s + (0.774 − 0.632i)16-s − 0.589·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.925 + 0.379i$
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: \((2,\ 312,\ (\ :3/2),\ 0.925 + 0.379i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4397022981\)
\(L(\frac12)\) \(\approx\) \(0.4397022981\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.481 + 2.78i)T \)
3 \( 1 + 3iT \)
13 \( 1 + (-29.0 - 36.7i)T \)
good5 \( 1 + 21.7T + 125T^{2} \)
7 \( 1 + 13.0iT - 343T^{2} \)
11 \( 1 + 21.2T + 1.33e3T^{2} \)
17 \( 1 + 41.3T + 4.91e3T^{2} \)
19 \( 1 + 147.T + 6.85e3T^{2} \)
23 \( 1 - 58.4T + 1.21e4T^{2} \)
29 \( 1 + 180. iT - 2.43e4T^{2} \)
31 \( 1 + 272. iT - 2.97e4T^{2} \)
37 \( 1 - 337.T + 5.06e4T^{2} \)
41 \( 1 - 118. iT - 6.89e4T^{2} \)
43 \( 1 - 404. iT - 7.95e4T^{2} \)
47 \( 1 + 145. iT - 1.03e5T^{2} \)
53 \( 1 - 409. iT - 1.48e5T^{2} \)
59 \( 1 + 809.T + 2.05e5T^{2} \)
61 \( 1 + 34.5iT - 2.26e5T^{2} \)
67 \( 1 - 750.T + 3.00e5T^{2} \)
71 \( 1 - 455. iT - 3.57e5T^{2} \)
73 \( 1 - 130. iT - 3.89e5T^{2} \)
79 \( 1 - 108.T + 4.93e5T^{2} \)
83 \( 1 - 201.T + 5.71e5T^{2} \)
89 \( 1 - 1.26e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.42e3iT - 9.12e5T^{2} \)
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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

−11.15334159485919115073263781517, −10.80723493082111981672845883509, −9.243485806986345309994501448607, −8.139768796318296277833415584427, −7.78683524682816828646249687568, −6.51728792368300507648040453162, −4.40423290589202957271020454453, −4.04236740160421471182801446546, −2.56556471827586503618069264899, −0.77145219759340355305041916036, 0.27351847978852472634049637569, 3.26428858097622082011614204305, 4.32516821699397004564654807688, 5.18976501687228516183380951464, 6.53143272936544919251780707498, 7.59758717778727223125400866878, 8.591403941141846203212703619887, 8.769502497267360231764709093122, 10.55367393163664547798018459353, 11.00734510654454756854399269380

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