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Properties

Label 2-2349-261.173-c0-0-2
Degree $2$
Conductor $2349$
Sign $0.984 + 0.173i$
Analytic cond. $1.17230$
Root an. cond. $1.08272$
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  + (0.766 − 1.32i)2-s + (−0.673 − 1.16i)4-s + (−0.766 + 1.32i)7-s − 0.532·8-s + (−0.939 + 1.62i)11-s + (0.939 + 1.62i)13-s + (1.17 + 2.03i)14-s + (0.266 − 0.460i)16-s − 0.347·17-s + (1.43 + 2.49i)22-s + (−0.5 + 0.866i)25-s + 2.87·26-s + 2.06·28-s + (0.5 − 0.866i)29-s + (−0.673 − 1.16i)32-s + ⋯
L(s)  = 1  + (0.766 − 1.32i)2-s + (−0.673 − 1.16i)4-s + (−0.766 + 1.32i)7-s − 0.532·8-s + (−0.939 + 1.62i)11-s + (0.939 + 1.62i)13-s + (1.17 + 2.03i)14-s + (0.266 − 0.460i)16-s − 0.347·17-s + (1.43 + 2.49i)22-s + (−0.5 + 0.866i)25-s + 2.87·26-s + 2.06·28-s + (0.5 − 0.866i)29-s + (−0.673 − 1.16i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(1.17230\)
Root analytic conductor: \(1.08272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (1565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2349,\ (\ :0),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.475935385\)
\(L(\frac12)\) \(\approx\) \(1.475935385\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + 0.347T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - 1.87T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.395893588811778483703814869134, −8.759286420956389941504758337940, −7.55977725045349814101855363196, −6.65457768034602351418044643430, −5.78640380481802281315315610697, −4.92780541359191845409740848931, −4.21951277882541392361703317203, −3.31576288073867391391122784879, −2.24043034729176087856268514637, −1.91281393945925911547618034110, 0.77742057119164821224225223869, 3.11224540317726819242230708729, 3.53711261621491819547173921498, 4.55738429453936263979707506134, 5.50749482809354794867410792001, 6.08407654132006912024079255563, 6.67660041618918204818146364973, 7.63281164843883532067911520851, 8.119031881840697887822757515954, 8.725663082369898596828236875230

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