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Properties

Label 2-2352-7.4-c1-0-0
Degree $2$
Conductor $2352$
Sign $0.386 - 0.922i$
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  + (0.5 − 0.866i)3-s + (−2.13 − 3.70i)5-s + (−0.499 − 0.866i)9-s + (−2.13 + 3.70i)11-s + 1.27·13-s − 4.27·15-s + (−2 + 3.46i)17-s + (−0.637 − 1.10i)19-s + (2 + 3.46i)23-s + (−6.63 + 11.4i)25-s − 0.999·27-s − 2.27·29-s + (0.5 − 0.866i)31-s + (2.13 + 3.70i)33-s + (−2.63 − 4.56i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.955 − 1.65i)5-s + (−0.166 − 0.288i)9-s + (−0.644 + 1.11i)11-s + 0.353·13-s − 1.10·15-s + (−0.485 + 0.840i)17-s + (−0.146 − 0.253i)19-s + (0.417 + 0.722i)23-s + (−1.32 + 2.29i)25-s − 0.192·27-s − 0.422·29-s + (0.0898 − 0.155i)31-s + (0.372 + 0.644i)33-s + (−0.433 − 0.751i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5526642749\)
\(L(\frac12)\) \(\approx\) \(0.5526642749\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (2.13 + 3.70i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.13 - 3.70i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.27T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.637 + 1.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.27T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.63 + 4.56i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 7.27T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.862 - 1.49i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.13 - 5.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.63 + 6.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (-1.63 + 2.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.77 + 3.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.274T + 83T^{2} \)
89 \( 1 + (-2.27 - 3.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−8.926641464282452934071438560310, −8.390251663771764225328802862344, −7.63776616012794034769228455287, −7.15270702423031648907000549874, −5.89608631285451916582971448154, −5.05152519120248694679734574939, −4.34492976942104251977218069923, −3.57783262419961320200628409693, −2.12285733087057537993755374554, −1.17463120821293284080853663218, 0.19304934816264362709072944297, 2.39802366218911424416466655087, 3.16067062382694654740118550749, 3.67930847594439919728008632825, 4.71324592271630888377744880141, 5.77516716760922360435652456157, 6.69806931151033070630169517884, 7.23120601772400711804630220462, 8.224771449173058797756799481852, 8.559294562087530454692788086491

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