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Properties

Label 2-42-7.4-c3-0-2
Degree $2$
Conductor $42$
Sign $0.701 - 0.712i$
Analytic cond. $2.47808$
Root an. cond. $1.57419$
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 + 1.73i)2-s + (1.5 − 2.59i)3-s + (−1.99 + 3.46i)4-s + (7.5 + 12.9i)5-s + 6·6-s + (17.5 − 6.06i)7-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (−15 + 25.9i)10-s + (4.5 − 7.79i)11-s + (6.00 + 10.3i)12-s − 88·13-s + (28 + 24.2i)14-s + 45·15-s + (−8 − 13.8i)16-s + (42 − 72.7i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.670 + 1.16i)5-s + 0.408·6-s + (0.944 − 0.327i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.474 + 0.821i)10-s + (0.123 − 0.213i)11-s + (0.144 + 0.249i)12-s − 1.87·13-s + (0.534 + 0.462i)14-s + 0.774·15-s + (−0.125 − 0.216i)16-s + (0.599 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42\)    =    \(2 \cdot 3 \cdot 7\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(2.47808\)
Root analytic conductor: \(1.57419\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{42} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 42,\ (\ :3/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.64138 + 0.687876i\)
\(L(\frac12)\) \(\approx\) \(1.64138 + 0.687876i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 + (-17.5 + 6.06i)T \)
good5 \( 1 + (-7.5 - 12.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-4.5 + 7.79i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 88T + 2.19e3T^{2} \)
17 \( 1 + (-42 + 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (52 + 90.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-42 - 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 51T + 2.43e4T^{2} \)
31 \( 1 + (92.5 - 160. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (22 + 38.1i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 168T + 6.89e4T^{2} \)
43 \( 1 - 326T + 7.95e4T^{2} \)
47 \( 1 + (-69 - 119. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (319.5 - 553. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (79.5 - 137. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (361 + 625. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-83 + 143. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.08e3T + 3.57e5T^{2} \)
73 \( 1 + (109 - 188. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-291.5 - 504. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 597T + 5.71e5T^{2} \)
89 \( 1 + (-519 - 898. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 169T + 9.12e5T^{2} \)
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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

−15.26178417696291290916343252557, −14.31271419521080606216729321281, −13.90090672773671881444591635387, −12.31646414784237188265789769665, −10.93250203824558419472067081365, −9.408694880818202920241562305098, −7.60629751575243240440379831064, −6.84023440561201575985433160898, −5.07631663123814385231326386770, −2.66587029477661293352619596820, 1.97503031701758614170387572369, 4.47319882898785407768263560191, 5.48980522042940511769570272467, 8.160171991430618535539227517499, 9.390360888492254227884243579686, 10.40802741112530223858397834119, 12.07834153877700984898817274917, 12.81602736417602124382330247547, 14.36934532298448455643604127093, 14.94142550864304259034091255737

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