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Properties

Label 2-2888-2888.339-c0-0-0
Degree $2$
Conductor $2888$
Sign $-0.973 + 0.227i$
Analytic cond. $1.44129$
Root an. cond. $1.20054$
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  + (0.811 − 0.584i)2-s + (−0.589 − 1.27i)3-s + (0.315 − 0.948i)4-s + (−1.22 − 0.692i)6-s + (−0.298 − 0.954i)8-s + (−0.637 + 0.746i)9-s + (1.25 − 1.52i)11-s + (−1.39 + 0.154i)12-s + (−0.800 − 0.599i)16-s + (1.95 + 0.400i)17-s + (−0.0810 + 0.978i)18-s + (−0.912 + 0.410i)19-s + (0.127 − 1.97i)22-s + (−1.04 + 0.943i)24-s + (0.690 − 0.723i)25-s + ⋯
L(s)  = 1  + (0.811 − 0.584i)2-s + (−0.589 − 1.27i)3-s + (0.315 − 0.948i)4-s + (−1.22 − 0.692i)6-s + (−0.298 − 0.954i)8-s + (−0.637 + 0.746i)9-s + (1.25 − 1.52i)11-s + (−1.39 + 0.154i)12-s + (−0.800 − 0.599i)16-s + (1.95 + 0.400i)17-s + (−0.0810 + 0.978i)18-s + (−0.912 + 0.410i)19-s + (0.127 − 1.97i)22-s + (−1.04 + 0.943i)24-s + (0.690 − 0.723i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.973 + 0.227i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.973 + 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.726969550\)
\(L(\frac12)\) \(\approx\) \(1.726969550\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.811 + 0.584i)T \)
19 \( 1 + (0.912 - 0.410i)T \)
good3 \( 1 + (0.589 + 1.27i)T + (-0.649 + 0.760i)T^{2} \)
5 \( 1 + (-0.690 + 0.723i)T^{2} \)
7 \( 1 + (-0.451 - 0.892i)T^{2} \)
11 \( 1 + (-1.25 + 1.52i)T + (-0.191 - 0.981i)T^{2} \)
13 \( 1 + (0.861 + 0.507i)T^{2} \)
17 \( 1 + (-1.95 - 0.400i)T + (0.919 + 0.393i)T^{2} \)
23 \( 1 + (-0.983 + 0.182i)T^{2} \)
29 \( 1 + (0.729 + 0.684i)T^{2} \)
31 \( 1 + (-0.0275 + 0.999i)T^{2} \)
37 \( 1 + (-0.945 - 0.324i)T^{2} \)
41 \( 1 + (1.02 - 1.63i)T + (-0.435 - 0.900i)T^{2} \)
43 \( 1 + (0.931 - 1.68i)T + (-0.531 - 0.847i)T^{2} \)
47 \( 1 + (-0.484 + 0.875i)T^{2} \)
53 \( 1 + (0.999 - 0.0183i)T^{2} \)
59 \( 1 + (1.12 + 0.0413i)T + (0.997 + 0.0734i)T^{2} \)
61 \( 1 + (0.263 + 0.964i)T^{2} \)
67 \( 1 + (-0.0306 + 0.666i)T + (-0.995 - 0.0917i)T^{2} \)
71 \( 1 + (0.263 - 0.964i)T^{2} \)
73 \( 1 + (0.705 - 0.795i)T + (-0.119 - 0.992i)T^{2} \)
79 \( 1 + (-0.999 + 0.0367i)T^{2} \)
83 \( 1 + (-1.45 - 0.0803i)T + (0.993 + 0.110i)T^{2} \)
89 \( 1 + (-0.599 - 0.675i)T + (-0.119 + 0.992i)T^{2} \)
97 \( 1 + (0.0159 + 0.346i)T + (-0.995 + 0.0917i)T^{2} \)
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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

−8.432714167900286097773364512691, −7.78722559475132632692808246459, −6.65552316096473483754990595386, −6.21258941443269050971146022776, −5.82276137422957693082123950904, −4.73901401509453261131862150477, −3.64404178837085913117103212738, −2.96955977740178421652890105893, −1.54673547596990976487126887146, −1.01404379459063432666554391674, 1.93038181770485710214829868289, 3.43259174392355942174292158828, 3.89102908202071456852014394188, 4.82824692095826169463413439357, 5.17791291898496368154964494708, 6.07269076379111091508921010739, 6.99861147751201495103807191950, 7.43688431775165299505995640580, 8.698141273684332273850368143735, 9.261963725235060981549042730011

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