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Properties

Label 2-2352-147.107-c0-0-0
Degree $2$
Conductor $2352$
Sign $0.967 - 0.253i$
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 + (0.440 − 1.92i)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 + (0.722 − 1.84i)39-s + (−1.19 − 1.49i)43-s + (0.0747 + 0.997i)49-s + (−0.914 + 1.14i)57-s + (−0.134 + 1.79i)61-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)3-s + (0.733 + 0.680i)7-s + (0.955 + 0.294i)9-s + (0.440 − 1.92i)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 + (0.722 − 1.84i)39-s + (−1.19 − 1.49i)43-s + (0.0747 + 0.997i)49-s + (−0.914 + 1.14i)57-s + (−0.134 + 1.79i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.967 - 0.253i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.967 - 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.847038169\)
\(L(\frac12)\) \(\approx\) \(1.847038169\)
\(L(1)\) 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 \( 1 + (-0.440 + 1.92i)T + (-0.900 - 0.433i)T^{2} \)
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 \( 1 + (1.19 + 1.49i)T + (-0.222 + 0.974i)T^{2} \)
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 \( 1 + (0.134 - 1.79i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (-0.365 - 0.632i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (1.21 - 1.12i)T + (0.0747 - 0.997i)T^{2} \)
79 \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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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

−8.942229662496635373293976839153, −8.425834745660845846115042480342, −7.890140882552185563146141604973, −7.19415088907652404484937794067, −5.80039923465728246133306949183, −5.42654567278748456830223587752, −4.19098268774099503487518003187, −3.44206275780468711881843252013, −2.48566991054796659615350535615, −1.53535039062202461880369649963, 1.45222423205320220948568015029, 2.23565466144448216190233748107, 3.43064038173837170086769614965, 4.42854425976628075060312425763, 4.69547087697128113346404175547, 6.51216555508543579924185307110, 6.71083159602797010802595082754, 7.78893025376073581103424529016, 8.325921676599316531518593439901, 9.083641884477387637027579647788

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