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Properties

Label 2-2352-588.479-c0-0-1
Degree $2$
Conductor $2352$
Sign $0.710 + 0.703i$
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  + (−0.988 + 0.149i)3-s + (0.733 − 0.680i)7-s + (0.955 − 0.294i)9-s + (0.290 − 0.0663i)13-s + (−0.733 − 1.26i)19-s + (−0.623 + 0.781i)21-s + (0.733 + 0.680i)25-s + (−0.900 + 0.433i)27-s + (−0.0747 + 0.129i)31-s + (−0.123 − 1.64i)37-s + (−0.277 + 0.108i)39-s + (−0.460 − 0.367i)43-s + (0.0747 − 0.997i)49-s + (0.914 + 1.14i)57-s + (0.865 − 0.0648i)61-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)3-s + (0.733 − 0.680i)7-s + (0.955 − 0.294i)9-s + (0.290 − 0.0663i)13-s + (−0.733 − 1.26i)19-s + (−0.623 + 0.781i)21-s + (0.733 + 0.680i)25-s + (−0.900 + 0.433i)27-s + (−0.0747 + 0.129i)31-s + (−0.123 − 1.64i)37-s + (−0.277 + 0.108i)39-s + (−0.460 − 0.367i)43-s + (0.0747 − 0.997i)49-s + (0.914 + 1.14i)57-s + (0.865 − 0.0648i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.710 + 0.703i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.710 + 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9157304065\)
\(L(\frac12)\) \(\approx\) \(0.9157304065\)
\(L(1)\) 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 \( 1 + (0.988 - 0.149i)T \)
7 \( 1 + (-0.733 + 0.680i)T \)
good5 \( 1 + (-0.733 - 0.680i)T^{2} \)
11 \( 1 + (0.826 + 0.563i)T^{2} \)
13 \( 1 + (-0.290 + 0.0663i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (0.365 - 0.930i)T^{2} \)
19 \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.365 + 0.930i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.123 + 1.64i)T + (-0.988 + 0.149i)T^{2} \)
41 \( 1 + (-0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.460 + 0.367i)T + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.988 + 0.149i)T^{2} \)
59 \( 1 + (0.733 - 0.680i)T^{2} \)
61 \( 1 + (-0.865 + 0.0648i)T + (0.988 - 0.149i)T^{2} \)
67 \( 1 + (-1.61 - 0.930i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.766 - 0.825i)T + (-0.0747 - 0.997i)T^{2} \)
79 \( 1 + (-1.72 + 0.997i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.826 - 0.563i)T^{2} \)
97 \( 1 + 1.94iT - T^{2} \)
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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

−9.081833111074718228443079818346, −8.338552191156021309321819104237, −7.21064968116067583493337571127, −6.93150557079570088467599658958, −5.86222180378496482202783209220, −5.07570775715640951045266082087, −4.42938932391172667955035027312, −3.57023420920226003850309830046, −2.05913513792703603061356779771, −0.808216166066214732928758217788, 1.31357817656027967580070081997, 2.30123226132333570460580317897, 3.73030929748524143827939200133, 4.71432371911970885755978617108, 5.30000037470373891571840053673, 6.22507878668781385399289732167, 6.66426677973582498355945826315, 7.899952624015753539793369098055, 8.285825945653542105660090208639, 9.291115083365419632969375671634

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