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Properties

Label 2-104-104.83-c1-0-1
Degree $2$
Conductor $104$
Sign $-0.289 - 0.957i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s − 3-s − 2i·4-s + (2.54 + 2.54i)5-s + (1 − i)6-s + (−2.54 + 2.54i)7-s + (2 + 2i)8-s − 2·9-s − 5.09·10-s + (1 + i)11-s + 2i·12-s + (2.54 + 2.54i)13-s − 5.09i·14-s + (−2.54 − 2.54i)15-s − 4·16-s − 3i·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 0.577·3-s − i·4-s + (1.14 + 1.14i)5-s + (0.408 − 0.408i)6-s + (−0.963 + 0.963i)7-s + (0.707 + 0.707i)8-s − 0.666·9-s − 1.61·10-s + (0.301 + 0.301i)11-s + 0.577i·12-s + (0.707 + 0.707i)13-s − 1.36i·14-s + (−0.658 − 0.658i)15-s − 16-s − 0.727i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ -0.289 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.382697 + 0.515726i\)
\(L(\frac12)\) \(\approx\) \(0.382697 + 0.515726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
13 \( 1 + (-2.54 - 2.54i)T \)
good3 \( 1 + T + 3T^{2} \)
5 \( 1 + (-2.54 - 2.54i)T + 5iT^{2} \)
7 \( 1 + (2.54 - 2.54i)T - 7iT^{2} \)
11 \( 1 + (-1 - i)T + 11iT^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + (-2 + 2i)T - 19iT^{2} \)
23 \( 1 - 5.09T + 23T^{2} \)
29 \( 1 + 5.09iT - 29T^{2} \)
31 \( 1 + (5.09 + 5.09i)T + 31iT^{2} \)
37 \( 1 + (2.54 - 2.54i)T - 37iT^{2} \)
41 \( 1 + (-6 + 6i)T - 41iT^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + (2.54 - 2.54i)T - 47iT^{2} \)
53 \( 1 - 5.09iT - 53T^{2} \)
59 \( 1 + (-8 - 8i)T + 59iT^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-3 + 3i)T - 67iT^{2} \)
71 \( 1 + (-7.64 - 7.64i)T + 71iT^{2} \)
73 \( 1 + (6 + 6i)T + 73iT^{2} \)
79 \( 1 + 5.09iT - 79T^{2} \)
83 \( 1 + (-5 + 5i)T - 83iT^{2} \)
89 \( 1 + (2 + 2i)T + 89iT^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−14.25846987770983420492053383154, −13.38253692476843098282926551077, −11.65571896803484622456948625666, −10.78968984458762246601037391337, −9.558937967384012206759723928695, −9.039021026010379074162872165601, −7.03501935581057867994513552218, −6.23515471476455278429499283065, −5.52985957166114308155843123738, −2.56799555875125013026098134348, 1.08173189395811207877629341027, 3.46072327223121112782592454106, 5.36789693522002550494786699393, 6.64000696166601176652651517608, 8.401852584101365089044823613797, 9.296557111346423641802637298505, 10.29775359764292976226402564992, 11.12279472756644437787749760808, 12.65207214052451607068231670570, 13.00369865335117237236138588059

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