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Properties

Label 1-3744-3744.979-r1-0-0
Degree $1$
Conductor $3744$
Sign $-0.164 + 0.986i$
Analytic cond. $402.348$
Root an. cond. $402.348$
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  + (−0.258 − 0.965i)5-s − i·7-s + (−0.258 − 0.965i)11-s + (0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s − i·23-s + (−0.866 + 0.5i)25-s + (0.965 − 0.258i)29-s + (−0.5 + 0.866i)31-s + (−0.965 + 0.258i)35-s + (0.965 + 0.258i)37-s − i·41-s + (0.707 − 0.707i)43-s + (0.5 + 0.866i)47-s − 49-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)5-s − i·7-s + (−0.258 − 0.965i)11-s + (0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s − i·23-s + (−0.866 + 0.5i)25-s + (0.965 − 0.258i)29-s + (−0.5 + 0.866i)31-s + (−0.965 + 0.258i)35-s + (0.965 + 0.258i)37-s − i·41-s + (0.707 − 0.707i)43-s + (0.5 + 0.866i)47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.164 + 0.986i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ -0.164 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5083254594 - 0.6002961875i\)
\(L(\frac12)\) \(\approx\) \(-0.5083254594 - 0.6002961875i\)
\(L(1)\) \(\approx\) \(0.7640850168 - 0.4558144474i\)
\(L(1)\) \(\approx\) \(0.7640850168 - 0.4558144474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.258 - 0.965i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.965 - 0.258i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.258 - 0.965i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 - T \)
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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

−18.953044527461237814173062205624, −18.185597425771992098366130722035, −17.79536293362175934324556935436, −16.98217093356757565597511134758, −16.04920179997435285072198356538, −15.24809239235094219008739878161, −14.99656985426303072888299414715, −14.40687145382735393086588637635, −13.36247262725317364432915012153, −12.64901847329253941003638063059, −12.06993259716317338305042901200, −11.28496220958568376112455295731, −10.680240806959052987242492085113, −9.852900357808672434286929114186, −9.31711990334066860718638884489, −8.24318113015501267734192513322, −7.77286811994648804532151247470, −6.88796331369840963692523615831, −6.18826493350668517952974148436, −5.55904412678483309089662719108, −4.54186935219565516884453322031, −3.818196947480012154225880097007, −2.81828712343444509247468562213, −2.31246050122781351044616466834, −1.40742038815877660513761811496, 0.156895103032857097706998523311, 0.68528131448080252847616150229, 1.50625024570346131273922589109, 2.70723093667687806422550013544, 3.54076903098877745323649038933, 4.391511712832929940292309057497, 4.86155375264099101963655816995, 5.84803872724320867597102682818, 6.54422236320216928995508747130, 7.56857863993843631746554560193, 8.03343133088818775253748413727, 8.83661440341991960327231838763, 9.42580551789151984205513137345, 10.565866100891596511169655513877, 10.75785648745961529925315143779, 11.840620093986142555330070669326, 12.463388699293942039252663678438, 13.08719544318790827753044217709, 13.97011677191195783502028454893, 14.189713007558202317542845606747, 15.38696066090355357451728837416, 16.103972326210626769418632416818, 16.555982190176961103551399191262, 17.094258118675056826077848029870, 17.82566652918463581595952285195

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