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Properties

Label 2-2352-588.419-c0-0-0
Degree $2$
Conductor $2352$
Sign $0.987 - 0.159i$
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  + (−0.222 + 0.974i)3-s + (0.900 − 0.433i)7-s + (−0.900 − 0.433i)9-s + (−0.846 − 1.75i)13-s + 1.80·19-s + (0.222 + 0.974i)21-s + (0.900 + 0.433i)25-s + (0.623 − 0.781i)27-s + 1.24·31-s + (−0.777 − 0.974i)37-s + (1.90 − 0.433i)39-s + (−0.846 + 0.193i)43-s + (0.623 − 0.781i)49-s + (−0.400 + 1.75i)57-s + (−1.22 + 0.974i)61-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)3-s + (0.900 − 0.433i)7-s + (−0.900 − 0.433i)9-s + (−0.846 − 1.75i)13-s + 1.80·19-s + (0.222 + 0.974i)21-s + (0.900 + 0.433i)25-s + (0.623 − 0.781i)27-s + 1.24·31-s + (−0.777 − 0.974i)37-s + (1.90 − 0.433i)39-s + (−0.846 + 0.193i)43-s + (0.623 − 0.781i)49-s + (−0.400 + 1.75i)57-s + (−1.22 + 0.974i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.987 - 0.159i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.987 - 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188348868\)
\(L(\frac12)\) \(\approx\) \(1.188348868\)
\(L(1)\) 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 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (-0.900 + 0.433i)T \)
good5 \( 1 + (-0.900 - 0.433i)T^{2} \)
11 \( 1 + (0.623 - 0.781i)T^{2} \)
13 \( 1 + (0.846 + 1.75i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (-0.222 + 0.974i)T^{2} \)
19 \( 1 - 1.80T + T^{2} \)
23 \( 1 + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 - 1.24T + T^{2} \)
37 \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \)
41 \( 1 + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.846 - 0.193i)T + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 + 0.974i)T^{2} \)
59 \( 1 + (0.900 - 0.433i)T^{2} \)
61 \( 1 + (1.22 - 0.974i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 - 1.94iT - T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.678 + 1.40i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 - 1.56iT - T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.623 + 0.781i)T^{2} \)
97 \( 1 - 0.867iT - T^{2} \)
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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

−9.300102520194128722404192588978, −8.403820807694711641165349784995, −7.74573480897822651957001987446, −7.01808486959174117050924220218, −5.66605249415156218462791631461, −5.21717065241600994451944995081, −4.56865613839137988650369929491, −3.43774355552937651077468017461, −2.75142387428864099272148810624, −0.983516527678718631536937445676, 1.32103635466629156961269509082, 2.16714845528144724723684232906, 3.16472997534402778831624087824, 4.74363588657728526154632572888, 5.04336177712577856605666700683, 6.21196518452297336687645665339, 6.87861856903058311171936988678, 7.55951090082626556366021794022, 8.285998221576989927724359561492, 9.025064404790248659966377453243

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