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Properties

Label 1-4024-4024.253-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.856 + 0.516i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (0.277 + 0.960i)3-s + (−0.838 + 0.544i)5-s + (0.731 + 0.682i)7-s + (−0.845 + 0.533i)9-s + (−0.580 − 0.814i)11-s + (0.649 + 0.760i)13-s + (−0.756 − 0.654i)15-s + (0.994 + 0.0999i)17-s + (0.205 − 0.978i)19-s + (−0.452 + 0.891i)21-s + (0.920 + 0.389i)23-s + (0.407 − 0.913i)25-s + (−0.747 − 0.663i)27-s + (0.739 − 0.673i)29-s + (0.539 + 0.842i)31-s + ⋯
L(s)  = 1  + (0.277 + 0.960i)3-s + (−0.838 + 0.544i)5-s + (0.731 + 0.682i)7-s + (−0.845 + 0.533i)9-s + (−0.580 − 0.814i)11-s + (0.649 + 0.760i)13-s + (−0.756 − 0.654i)15-s + (0.994 + 0.0999i)17-s + (0.205 − 0.978i)19-s + (−0.452 + 0.891i)21-s + (0.920 + 0.389i)23-s + (0.407 − 0.913i)25-s + (−0.747 − 0.663i)27-s + (0.739 − 0.673i)29-s + (0.539 + 0.842i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.856 + 0.516i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.856 + 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4265320871 + 1.533708483i\)
\(L(\frac12)\) \(\approx\) \(0.4265320871 + 1.533708483i\)
\(L(1)\) \(\approx\) \(0.9157054503 + 0.5893577369i\)
\(L(1)\) \(\approx\) \(0.9157054503 + 0.5893577369i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.277 + 0.960i)T \)
5 \( 1 + (-0.838 + 0.544i)T \)
7 \( 1 + (0.731 + 0.682i)T \)
11 \( 1 + (-0.580 - 0.814i)T \)
13 \( 1 + (0.649 + 0.760i)T \)
17 \( 1 + (0.994 + 0.0999i)T \)
19 \( 1 + (0.205 - 0.978i)T \)
23 \( 1 + (0.920 + 0.389i)T \)
29 \( 1 + (0.739 - 0.673i)T \)
31 \( 1 + (0.539 + 0.842i)T \)
37 \( 1 + (-0.999 - 0.0250i)T \)
41 \( 1 + (-0.977 + 0.211i)T \)
43 \( 1 + (-0.955 + 0.295i)T \)
47 \( 1 + (-0.965 + 0.259i)T \)
53 \( 1 + (0.205 + 0.978i)T \)
59 \( 1 + (0.372 + 0.928i)T \)
61 \( 1 + (-0.620 - 0.784i)T \)
67 \( 1 + (0.756 - 0.654i)T \)
71 \( 1 + (0.889 + 0.457i)T \)
73 \( 1 + (-0.590 + 0.806i)T \)
79 \( 1 + (0.384 + 0.923i)T \)
83 \( 1 + (0.229 + 0.973i)T \)
89 \( 1 + (0.974 - 0.223i)T \)
97 \( 1 + (0.998 - 0.0500i)T \)
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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

−18.31384926040880245641579912217, −17.55124117794325243865589384736, −16.93528715310370037869333097368, −16.24969578377390306441429397336, −15.31432134517218688049069615744, −14.774633752971431176184765165898, −14.10156335429211959900398708841, −13.20110360747874586639424395952, −12.84388312186091494546087977417, −11.96493149950578007965143050378, −11.64349210610590699432372666173, −10.56938376407262864643588951832, −10.055535787929663186858217433195, −8.79359651626714451973927874904, −8.23795491177019831391867254557, −7.75997043761590830217950619344, −7.22817508384735460087042686661, −6.38725731372582747892991612011, −5.21163093469405868694516816087, −4.89936393278078330381061777814, −3.608994047993292653248539688, −3.26689412317942732683645369079, −1.95824757998041205547342275600, −1.265544124894852776260602404731, −0.50450466131916626065464494618, 1.07413383743689164593782310118, 2.34332842812425428525764603709, 3.1710583695312216347784578702, 3.50906650785312399343358589512, 4.68605452753702451100565741517, 5.0308148823754088420324147815, 5.96985031643339917632279721820, 6.85172933626350556159284260928, 7.83210872619738750032808359361, 8.46168414561163971396885090418, 8.80981419890841563171184719629, 9.832685431049038648436991546622, 10.57991016157864070829691914724, 11.32004121540933368897316827211, 11.50944422833399130028216884116, 12.39746094832699829744353524003, 13.67669426823916910547914031028, 14.00780691651958281137465508961, 14.84907011524260300151470422015, 15.46010889443559814885992837821, 15.79144766071801466365547832293, 16.543948643942079581598184996615, 17.29695559827165036039084241074, 18.23943421297556573181423014166, 18.83503527923145813824380996633

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