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Properties

Label 1-4033-4033.2008-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.977 - 0.209i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.977 - 0.209i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2008, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.977 - 0.209i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003289739047 + 0.03105838888i\)
\(L(\frac12)\) \(\approx\) \(0.003289739047 + 0.03105838888i\)
\(L(1)\) \(\approx\) \(0.7982297567 + 0.1082005705i\)
\(L(1)\) \(\approx\) \(0.7982297567 + 0.1082005705i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.245333207891986786323087890462, −17.757729509447306045003040603885, −16.41786346735504575179266751864, −15.97635202714836012329922387470, −15.36643092862909605629858723124, −14.62153079163693216728146117111, −14.17358418533944944376812194041, −13.29544308359801185528595259854, −12.228039844867962766335843803139, −11.868338599809463134389374691116, −11.33188387667532564141604802807, −10.55452100339183137638599125327, −10.06656111109160081804645402867, −9.33851549233731136215324837321, −8.6525421830085834438421897292, −7.675935763619819784305328906913, −6.38430857368257655532626975046, −5.85346644531642787710063844585, −5.49026897801696094788917738132, −4.330966069592473333917507976655, −3.56899122322371389104903695571, −3.205236579058704008864506547036, −2.50606182208097439639492465203, −1.10606488121579910110042751906, −0.00950878003848434240917444289, 1.10767388879917475507294649749, 1.96883863813347818415436391784, 3.3538886154780027096778525150, 4.15188081662551065794068275624, 4.56673942649445582859275083982, 5.59166329664766226656046673895, 6.19274154062521956573505476559, 6.98500673827094359358892379441, 7.48369971632122089277739876958, 8.03016118490675172914696518432, 9.02819454151409808504298389742, 9.514930779309606138160322016546, 10.79764366145675499665189026190, 11.560871218603105380040608207265, 12.306772222945133112624027032561, 12.69925517447104181564867390750, 13.32376497178067006435049248405, 14.00450122045510205719098980712, 14.59198097648609925600305405805, 15.68053066103445575190348592546, 16.23797334976110687704069372281, 16.70585796128019725980791721310, 17.41818864494796632063311075007, 17.72391724112283434481083019164, 18.87051585116402345417440530439

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