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Properties

Label 2-2352-147.137-c0-0-0
Degree $2$
Conductor $2352$
Sign $0.910 + 0.414i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
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  + (−0.826 + 0.563i)3-s + (0.988 + 0.149i)7-s + (0.365 − 0.930i)9-s + (1.03 − 1.29i)13-s + (−0.988 − 1.71i)19-s + (−0.900 + 0.433i)21-s + (−0.988 + 0.149i)25-s + (0.222 + 0.974i)27-s + (0.955 − 1.65i)31-s + (−1.40 + 0.432i)37-s + (−0.123 + 1.64i)39-s + (0.658 − 0.317i)43-s + (0.955 + 0.294i)49-s + (1.78 + 0.858i)57-s + (−0.425 + 0.131i)61-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)3-s + (0.988 + 0.149i)7-s + (0.365 − 0.930i)9-s + (1.03 − 1.29i)13-s + (−0.988 − 1.71i)19-s + (−0.900 + 0.433i)21-s + (−0.988 + 0.149i)25-s + (0.222 + 0.974i)27-s + (0.955 − 1.65i)31-s + (−1.40 + 0.432i)37-s + (−0.123 + 1.64i)39-s + (0.658 − 0.317i)43-s + (0.955 + 0.294i)49-s + (1.78 + 0.858i)57-s + (−0.425 + 0.131i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.910 + 0.414i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.910 + 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9902986880\)
\(L(\frac12)\) \(\approx\) \(0.9902986880\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.826 - 0.563i)T \)
7 \( 1 + (-0.988 - 0.149i)T \)
good5 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (0.733 - 0.680i)T^{2} \)
13 \( 1 + (-1.03 + 1.29i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.0747 - 0.997i)T^{2} \)
19 \( 1 + (0.988 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.955 + 1.65i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.40 - 0.432i)T + (0.826 - 0.563i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.658 + 0.317i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.955 - 0.294i)T^{2} \)
53 \( 1 + (-0.826 - 0.563i)T^{2} \)
59 \( 1 + (0.988 + 0.149i)T^{2} \)
61 \( 1 + (0.425 - 0.131i)T + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-1.44 + 0.218i)T + (0.955 - 0.294i)T^{2} \)
79 \( 1 + (-0.955 - 1.65i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−9.092059517077475905075472456577, −8.405346517088265625239681305409, −7.65992715617249610972379202946, −6.60602274146204089852071055597, −5.88470601268128234290161075042, −5.17319952954941135569152579424, −4.43847633907524628944300882769, −3.58929674245981148626943761216, −2.30763411432016953646803804732, −0.826980987394034488163265169230, 1.44578204741789146594300336549, 1.98359248044605318829658378785, 3.75941069371800295266570321975, 4.47504993019369767390237853387, 5.37071916659579516108962142669, 6.21452596985345852524993240923, 6.73726940108936436985948859317, 7.74786261588409208619330898066, 8.294585336620222993905653230708, 9.051710860607140867385461896413

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