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Properties

Label 2-2352-28.3-c1-0-29
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (3 − 1.73i)5-s + (−0.499 − 0.866i)9-s + (3 + 1.73i)11-s − 5.19i·13-s + 3.46i·15-s + (6 + 3.46i)17-s + (−3.5 − 6.06i)19-s + (3.5 − 6.06i)25-s + 0.999·27-s + (−2.5 + 4.33i)31-s + (−3 + 1.73i)33-s + (−0.5 − 0.866i)37-s + (4.5 + 2.59i)39-s − 10.3i·41-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (1.34 − 0.774i)5-s + (−0.166 − 0.288i)9-s + (0.904 + 0.522i)11-s − 1.44i·13-s + 0.894i·15-s + (1.45 + 0.840i)17-s + (−0.802 − 1.39i)19-s + (0.700 − 1.21i)25-s + 0.192·27-s + (−0.449 + 0.777i)31-s + (−0.522 + 0.301i)33-s + (−0.0821 − 0.142i)37-s + (0.720 + 0.416i)39-s − 1.62i·41-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: $\chi_{2352} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.229970676\)
\(L(\frac12)\) \(\approx\) \(2.229970676\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 + 1.73i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
17 \( 1 + (-6 - 3.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.5 + 7.79i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−9.063612335781129426339583293787, −8.402476117195128610810920786112, −7.31937147319486554677172043508, −6.34298678050721588464292886384, −5.62479475580597727560957611375, −5.15536397300438695115661271921, −4.19753776495038422714554256498, −3.13795774454789076896795800691, −1.91986960575059994913610143153, −0.863161756554439358181183586219, 1.32906963022462873338354535929, 2.06251900713278261862893669241, 3.15220864268317864617552025386, 4.18669324297706542238677364867, 5.44102065952285412942400214704, 6.09697011478991417040065035668, 6.55240958564517376145313109432, 7.31702374789954103649794288886, 8.264499095423831681303709712487, 9.252450178627002607378927750841

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