Properties

Label 2-43758-1.1-c1-0-4
Degree $2$
Conductor $43758$
Sign $1$
Analytic cond. $349.409$
Root an. cond. $18.6924$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·5-s − 8-s − 2·10-s − 11-s + 13-s + 16-s + 17-s + 8·19-s + 2·20-s + 22-s + 4·23-s − 25-s − 26-s − 6·29-s − 4·31-s − 32-s − 34-s − 6·37-s − 8·38-s − 2·40-s + 6·41-s + 12·43-s − 44-s − 4·46-s − 8·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s − 0.301·11-s + 0.277·13-s + 1/4·16-s + 0.242·17-s + 1.83·19-s + 0.447·20-s + 0.213·22-s + 0.834·23-s − 1/5·25-s − 0.196·26-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.171·34-s − 0.986·37-s − 1.29·38-s − 0.316·40-s + 0.937·41-s + 1.82·43-s − 0.150·44-s − 0.589·46-s − 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43758\)    =    \(2 \cdot 3^{2} \cdot 11 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(349.409\)
Root analytic conductor: \(18.6924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{43758} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 43758,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.013396500\)
\(L(\frac12)\) \(\approx\) \(2.013396500\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.56886629616435, −14.24039396315483, −13.64023326945026, −13.13304959932816, −12.66704946513430, −12.01450422302624, −11.42736986367832, −10.90613952908956, −10.55005025194665, −9.689819193865175, −9.454178708100571, −9.170606828978754, −8.289052799388903, −7.767775064376316, −7.250952871919228, −6.764020147399893, −5.880110036218543, −5.595812104383902, −5.076398985667189, −4.128648110626256, −3.264419767364599, −2.876354079199929, −1.910803786718398, −1.454047538105243, −0.5826479551707481, 0.5826479551707481, 1.454047538105243, 1.910803786718398, 2.876354079199929, 3.264419767364599, 4.128648110626256, 5.076398985667189, 5.595812104383902, 5.880110036218543, 6.764020147399893, 7.250952871919228, 7.767775064376316, 8.289052799388903, 9.170606828978754, 9.454178708100571, 9.689819193865175, 10.55005025194665, 10.90613952908956, 11.42736986367832, 12.01450422302624, 12.66704946513430, 13.13304959932816, 13.64023326945026, 14.24039396315483, 14.56886629616435

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