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Properties

Label 2-3703-161.104-c0-0-4
Degree $2$
Conductor $3703$
Sign $0.603 - 0.797i$
Analytic cond. $1.84803$
Root an. cond. $1.35942$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.698 − 0.449i)2-s + (−0.128 + 0.281i)4-s + (0.654 + 0.755i)7-s + (0.154 + 1.07i)8-s + (−0.959 − 0.281i)9-s + (0.239 + 0.153i)11-s + (0.797 + 0.234i)14-s + (0.389 + 0.449i)16-s + (−0.797 + 0.234i)18-s + 0.236·22-s + (0.841 − 0.540i)25-s + (−0.297 + 0.0872i)28-s + (0.698 + 1.53i)29-s + (−0.570 − 0.167i)32-s + (0.202 − 0.234i)36-s + (−1.25 − 0.368i)37-s + ⋯
L(s)  = 1  + (0.698 − 0.449i)2-s + (−0.128 + 0.281i)4-s + (0.654 + 0.755i)7-s + (0.154 + 1.07i)8-s + (−0.959 − 0.281i)9-s + (0.239 + 0.153i)11-s + (0.797 + 0.234i)14-s + (0.389 + 0.449i)16-s + (−0.797 + 0.234i)18-s + 0.236·22-s + (0.841 − 0.540i)25-s + (−0.297 + 0.0872i)28-s + (0.698 + 1.53i)29-s + (−0.570 − 0.167i)32-s + (0.202 − 0.234i)36-s + (−1.25 − 0.368i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3703\)    =    \(7 \cdot 23^{2}\)
Sign: $0.603 - 0.797i$
Analytic conductor: \(1.84803\)
Root analytic conductor: \(1.35942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3703} (1392, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3703,\ (\ :0),\ 0.603 - 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.688772227\)
\(L(\frac12)\) \(\approx\) \(1.688772227\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 \)
good2 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
3 \( 1 + (0.959 + 0.281i)T^{2} \)
5 \( 1 + (-0.841 + 0.540i)T^{2} \)
11 \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.142 + 0.989i)T^{2} \)
17 \( 1 + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.959 - 0.281i)T^{2} \)
37 \( 1 + (1.25 + 0.368i)T + (0.841 + 0.540i)T^{2} \)
41 \( 1 + (-0.841 + 0.540i)T^{2} \)
43 \( 1 + (0.273 - 1.89i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1.10 - 1.27i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.959 - 0.281i)T^{2} \)
67 \( 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + (-0.841 + 0.540i)T^{2} \)
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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

−8.702350685523861831898261973112, −8.318598878165350489164813778397, −7.39595699781263722137170186839, −6.41570750332073712616709637791, −5.56938628032187843644580475167, −4.97304460776199081005076346665, −4.27163419994393519360269533823, −3.15479780695226824743680501601, −2.72035433276439038458114266422, −1.58870180836410140039238221455, 0.801053176830192113565491968779, 2.09845574106582462391295239088, 3.39598790496021404620966804628, 4.08373899483667326298675608507, 5.02656655502578780356207087864, 5.37782849012031554411862724686, 6.35247704947148116322248291914, 6.93472878378162833666245196309, 7.76318505263070530218111071517, 8.535926452707160811557331325152

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