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Properties

Label 2-2352-7.2-c1-0-21
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 + (−1 + 1.73i)5-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s − 13-s + 1.99·15-s + (−0.5 + 0.866i)19-s + (0.500 + 0.866i)25-s + 0.999·27-s + 4·29-s + (−4.5 − 7.79i)31-s + (−0.999 + 1.73i)33-s + (−1.5 + 2.59i)37-s + (0.5 + 0.866i)39-s + 10·41-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.447 + 0.774i)5-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s − 0.277·13-s + 0.516·15-s + (−0.114 + 0.198i)19-s + (0.100 + 0.173i)25-s + 0.192·27-s + 0.742·29-s + (−0.808 − 1.39i)31-s + (−0.174 + 0.301i)33-s + (−0.246 + 0.427i)37-s + (0.0800 + 0.138i)39-s + 1.56·41-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: $\chi_{2352} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.067016867\)
\(L(\frac12)\) \(\approx\) \(1.067016867\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-8 + 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−8.684889161753761560473567019582, −7.909045691144720300120806875039, −7.32261834993700179439043149342, −6.57016608492475979686698495591, −5.84026189614859112568099672593, −4.96091405593089452204061687943, −3.84863333838841410091079739919, −2.99143157666048655603679958200, −2.01089182619269223536061129716, −0.47028297774634751305130093318, 0.958279965519298771505917870043, 2.40551943175937956168835354156, 3.55664990711288053508099302551, 4.52461183979568642344119666214, 4.95337912383628265493927035745, 5.86586756947129023559217404769, 6.85522947369371698647412011586, 7.63675229315057595033968778765, 8.480736461094823191957060942817, 9.095650843607342341286807058191

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