Properties

Label 2-2352-12.11-c1-0-57
Degree $2$
Conductor $2352$
Sign $-0.329 + 0.944i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  + (−1.70 + 0.323i)3-s − 1.55i·5-s + (2.79 − 1.09i)9-s + 2.53·13-s + (0.502 + 2.64i)15-s − 4.33i·17-s − 6.83i·19-s − 3.80·23-s + 2.58·25-s + (−4.39 + 2.77i)27-s + 8.33i·29-s + 4.89i·31-s + 1.58·37-s + (−4.31 + 0.818i)39-s − 7.44i·41-s + ⋯
L(s)  = 1  + (−0.982 + 0.186i)3-s − 0.695i·5-s + (0.930 − 0.366i)9-s + 0.702·13-s + (0.129 + 0.683i)15-s − 1.05i·17-s − 1.56i·19-s − 0.794·23-s + 0.516·25-s + (−0.845 + 0.533i)27-s + 1.54i·29-s + 0.879i·31-s + 0.260·37-s + (−0.690 + 0.131i)39-s − 1.16i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.329 + 0.944i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9634290630\)
\(L(\frac12)\) \(\approx\) \(0.9634290630\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.70 - 0.323i)T \)
7 \( 1 \)
good5 \( 1 + 1.55iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2.53T + 13T^{2} \)
17 \( 1 + 4.33iT - 17T^{2} \)
19 \( 1 + 6.83iT - 19T^{2} \)
23 \( 1 + 3.80T + 23T^{2} \)
29 \( 1 - 8.33iT - 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 - 1.58T + 37T^{2} \)
41 \( 1 + 7.44iT - 41T^{2} \)
43 \( 1 - 6.20iT - 43T^{2} \)
47 \( 1 - 5.38T + 47T^{2} \)
53 \( 1 + 10.5iT - 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + 8.19T + 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + 6.92iT - 79T^{2} \)
83 \( 1 + 4.82T + 83T^{2} \)
89 \( 1 + 4.33iT - 89T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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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

−9.072251949445506239629457776359, −7.972406034965537610737194577345, −6.96208068105549462677667867953, −6.50893485260412920927300887955, −5.32281639548588826334126583538, −5.00153078949759572348679606313, −4.11109752176415412283248723403, −2.98300108660803064722114112371, −1.47157019654962619058528487150, −0.42421961734103385154340242545, 1.24321048596786767867957207934, 2.33131988958023012541209995967, 3.74585175542072722046361198366, 4.28540018542469691784965079655, 5.61724194809316227325828466073, 6.11055360472435817905321049112, 6.63115373271758509059343407035, 7.80623857857204463403412399730, 8.083307220679887463414182256597, 9.422995215452806275120011598179

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