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Properties

Label 2-2352-588.131-c0-0-0
Degree $2$
Conductor $2352$
Sign $0.995 - 0.0960i$
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  + (−0.0747 − 0.997i)3-s + (0.365 + 0.930i)7-s + (−0.988 + 0.149i)9-s + (1.55 + 1.24i)13-s + (−0.365 + 0.632i)19-s + (0.900 − 0.433i)21-s + (−0.365 + 0.930i)25-s + (0.222 + 0.974i)27-s + (−0.733 − 1.26i)31-s + (1.40 + 1.29i)37-s + (1.12 − 1.64i)39-s + (0.129 + 0.268i)43-s + (−0.733 + 0.680i)49-s + (0.658 + 0.317i)57-s + (1.32 − 1.42i)61-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)3-s + (0.365 + 0.930i)7-s + (−0.988 + 0.149i)9-s + (1.55 + 1.24i)13-s + (−0.365 + 0.632i)19-s + (0.900 − 0.433i)21-s + (−0.365 + 0.930i)25-s + (0.222 + 0.974i)27-s + (−0.733 − 1.26i)31-s + (1.40 + 1.29i)37-s + (1.12 − 1.64i)39-s + (0.129 + 0.268i)43-s + (−0.733 + 0.680i)49-s + (0.658 + 0.317i)57-s + (1.32 − 1.42i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.995 - 0.0960i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.995 - 0.0960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.201519300\)
\(L(\frac12)\) \(\approx\) \(1.201519300\)
\(L(1)\) 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 \( 1 + (0.0747 + 0.997i)T \)
7 \( 1 + (-0.365 - 0.930i)T \)
good5 \( 1 + (0.365 - 0.930i)T^{2} \)
11 \( 1 + (0.955 + 0.294i)T^{2} \)
13 \( 1 + (-1.55 - 1.24i)T + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.826 - 0.563i)T^{2} \)
19 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.826 + 0.563i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.40 - 1.29i)T + (0.0747 + 0.997i)T^{2} \)
41 \( 1 + (0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.129 - 0.268i)T + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.733 - 0.680i)T^{2} \)
53 \( 1 + (-0.0747 + 0.997i)T^{2} \)
59 \( 1 + (-0.365 - 0.930i)T^{2} \)
61 \( 1 + (-1.32 + 1.42i)T + (-0.0747 - 0.997i)T^{2} \)
67 \( 1 + (-0.975 + 0.563i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.548 + 0.215i)T + (0.733 + 0.680i)T^{2} \)
79 \( 1 + (1.17 + 0.680i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.955 - 0.294i)T^{2} \)
97 \( 1 + 1.56iT - T^{2} \)
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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

−9.008993645208549521377150953934, −8.336135658810567062941441277639, −7.78538640250188705288092586806, −6.73556753863170882185649213836, −6.10224929689768634199589096837, −5.57002130803790264701343141676, −4.36755157267045806992971535226, −3.33585577259520151545764916482, −2.13410371155600160100396561580, −1.45517705064671732413871003382, 0.922406693256429764080668415927, 2.62550695556913035409182859164, 3.70931437018607797475832413791, 4.15165539444424034719075942233, 5.20033771616219475479925496357, 5.87245781321524210460075170343, 6.77638772647952413920295204925, 7.82602785107495850989079437457, 8.460638004736058605206402887227, 9.102652573111531332596180037475

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