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Properties

Label 2-2352-28.19-c1-0-39
Degree $2$
Conductor $2352$
Sign $-0.832 + 0.553i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (1.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s + (−4.5 + 2.59i)11-s − 6.92i·13-s + 1.73i·15-s + (−3 + 1.73i)17-s + (−1 + 1.73i)19-s + (−6 − 3.46i)23-s + (−1 − 1.73i)25-s − 0.999·27-s − 9·29-s + (−0.5 − 0.866i)31-s + (−4.5 − 2.59i)33-s + (1 − 1.73i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.670 + 0.387i)5-s + (−0.166 + 0.288i)9-s + (−1.35 + 0.783i)11-s − 1.92i·13-s + 0.447i·15-s + (−0.727 + 0.420i)17-s + (−0.229 + 0.397i)19-s + (−1.25 − 0.722i)23-s + (−0.200 − 0.346i)25-s − 0.192·27-s − 1.67·29-s + (−0.0898 − 0.155i)31-s + (−0.783 − 0.452i)33-s + (0.164 − 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.92iT - 13T^{2} \)
17 \( 1 + (3 - 1.73i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6 + 3.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (6 - 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 15T + 83T^{2} \)
89 \( 1 + (-9 - 5.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.66iT - 97T^{2} \)
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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

−8.522180689475615128259910096585, −7.998641953167718849923992342680, −7.31333857205856821317683222083, −6.07227444012463748695736273123, −5.60714790275426967615553097236, −4.71023288072853336325559763163, −3.72450114350390533127955574999, −2.65659094837945419488790906615, −2.07034521889647784913726846582, 0, 1.74628249524579255042806993058, 2.28850319238323413394620308373, 3.53886246705890993371970353102, 4.55352588469285104700919427414, 5.51025702861163891238816903934, 6.12897078131592627998769503819, 7.06849335411307126789252820580, 7.71243984824943506158696406428, 8.652135713560738278686154003425

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