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Properties

Label 2-102-51.47-c2-0-1
Degree $2$
Conductor $102$
Sign $-0.251 - 0.967i$
Analytic cond. $2.77929$
Root an. cond. $1.66712$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + (2.98 + 0.275i)3-s + 2.00·4-s + (−7.00 + 7.00i)5-s + (−4.22 − 0.388i)6-s + (−5.33 + 5.33i)7-s − 2.82·8-s + (8.84 + 1.64i)9-s + (9.91 − 9.91i)10-s + (−3.04 − 3.04i)11-s + (5.97 + 0.550i)12-s + 0.439·13-s + (7.54 − 7.54i)14-s + (−22.8 + 19.0i)15-s + 4.00·16-s + (14.3 + 9.05i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.995 + 0.0916i)3-s + 0.500·4-s + (−1.40 + 1.40i)5-s + (−0.704 − 0.0648i)6-s + (−0.762 + 0.762i)7-s − 0.353·8-s + (0.983 + 0.182i)9-s + (0.991 − 0.991i)10-s + (−0.277 − 0.277i)11-s + (0.497 + 0.0458i)12-s + 0.0338·13-s + (0.539 − 0.539i)14-s + (−1.52 + 1.26i)15-s + 0.250·16-s + (0.846 + 0.532i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 - 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $-0.251 - 0.967i$
Analytic conductor: \(2.77929\)
Root analytic conductor: \(1.66712\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{102} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :1),\ -0.251 - 0.967i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.552867 + 0.714853i\)
\(L(\frac12)\) \(\approx\) \(0.552867 + 0.714853i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (-2.98 - 0.275i)T \)
17 \( 1 + (-14.3 - 9.05i)T \)
good5 \( 1 + (7.00 - 7.00i)T - 25iT^{2} \)
7 \( 1 + (5.33 - 5.33i)T - 49iT^{2} \)
11 \( 1 + (3.04 + 3.04i)T + 121iT^{2} \)
13 \( 1 - 0.439T + 169T^{2} \)
19 \( 1 - 13.8iT - 361T^{2} \)
23 \( 1 + (-14.8 - 14.8i)T + 529iT^{2} \)
29 \( 1 + (-11.2 + 11.2i)T - 841iT^{2} \)
31 \( 1 + (20.9 + 20.9i)T + 961iT^{2} \)
37 \( 1 + (-44.8 - 44.8i)T + 1.36e3iT^{2} \)
41 \( 1 + (-14.5 - 14.5i)T + 1.68e3iT^{2} \)
43 \( 1 + 35.3iT - 1.84e3T^{2} \)
47 \( 1 + 20.6iT - 2.20e3T^{2} \)
53 \( 1 + 69.7T + 2.80e3T^{2} \)
59 \( 1 + 10.1T + 3.48e3T^{2} \)
61 \( 1 + (38.2 - 38.2i)T - 3.72e3iT^{2} \)
67 \( 1 - 52.7T + 4.48e3T^{2} \)
71 \( 1 + (-82.3 + 82.3i)T - 5.04e3iT^{2} \)
73 \( 1 + (-32.3 - 32.3i)T + 5.32e3iT^{2} \)
79 \( 1 + (43.5 - 43.5i)T - 6.24e3iT^{2} \)
83 \( 1 + 51.9T + 6.88e3T^{2} \)
89 \( 1 - 54.7iT - 7.92e3T^{2} \)
97 \( 1 + (-54.3 - 54.3i)T + 9.40e3iT^{2} \)
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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

−14.24968814227911180560513504485, −12.70070252769922290305257233609, −11.63078009095724347539759334824, −10.52790876027908933329333603764, −9.573107883749392947934654612923, −8.208382052864487381280326686402, −7.58477637539585977325794420079, −6.35539334310089421611075166556, −3.65071943183084917662866375188, −2.78614883241642827841440630274, 0.791074225604144908204824386532, 3.33252962090806440804472122370, 4.64496566846480429252977804833, 7.13777011650591995837928525651, 7.83473128756949605778825474480, 8.867550830313148388158930785450, 9.647308936615055962562448833474, 11.08299153069048014146926395939, 12.53072019213729045818772744766, 12.92582282974464596590291347098

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