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Properties

Label 1-3332-3332.1563-r1-0-0
Degree $1$
Conductor $3332$
Sign $-0.385 - 0.922i$
Analytic cond. $358.073$
Root an. cond. $358.073$
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  + (0.365 + 0.930i)3-s + (0.988 + 0.149i)5-s + (−0.733 + 0.680i)9-s + (−0.733 − 0.680i)11-s + (−0.222 + 0.974i)13-s + (0.222 + 0.974i)15-s + (0.5 + 0.866i)19-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)27-s + (0.900 − 0.433i)29-s + (−0.5 + 0.866i)31-s + (0.365 − 0.930i)33-s + (−0.826 − 0.563i)37-s + (−0.988 + 0.149i)39-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)3-s + (0.988 + 0.149i)5-s + (−0.733 + 0.680i)9-s + (−0.733 − 0.680i)11-s + (−0.222 + 0.974i)13-s + (0.222 + 0.974i)15-s + (0.5 + 0.866i)19-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)27-s + (0.900 − 0.433i)29-s + (−0.5 + 0.866i)31-s + (0.365 − 0.930i)33-s + (−0.826 − 0.563i)37-s + (−0.988 + 0.149i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.385 - 0.922i$
Analytic conductor: \(358.073\)
Root analytic conductor: \(358.073\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3332,\ (1:\ ),\ -0.385 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3016965410 + 0.4528289822i\)
\(L(\frac12)\) \(\approx\) \(-0.3016965410 + 0.4528289822i\)
\(L(1)\) \(\approx\) \(1.032089737 + 0.5129425302i\)
\(L(1)\) \(\approx\) \(1.032089737 + 0.5129425302i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.365 + 0.930i)T \)
5 \( 1 + (0.988 + 0.149i)T \)
11 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.826 - 0.563i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (-0.955 + 0.294i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (0.988 - 0.149i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
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

−18.10765399448146907093242878933, −17.70829861051229834093763537875, −17.09866715822830961555272842266, −16.18569114955564432751078084954, −15.19940478003994110565351288708, −14.71785976298509168415953188954, −13.84308766123423698411450268627, −13.27174894174032752455469257692, −12.78430076272163705156442461847, −12.17633608088381676020377353080, −11.226330407545264749229639838756, −10.21281914050887863147615136448, −9.86852544404996702114876568578, −8.73855606637009702604013085274, −8.38781809232713751829414999628, −7.28686494199406492257478173299, −6.921964445367529054456306917858, −5.93222007216742144662122291869, −5.31773463778907929686318770012, −4.54949685905698420088135100480, −3.082902674097969240378076263983, −2.65524752072702227255166247026, −1.85537286738506847890547808072, −0.97854315346159697046374479270, −0.07396284921813869050092885681, 1.45565707517471638125037119953, 2.2022264891893922329314293298, 3.13121354966356956598984133049, 3.67085930232304934562940758134, 4.85400036412496578106631343672, 5.31521038691633222875876623836, 6.06593739778973724955728563910, 6.95639781850925050599881250125, 7.94714130735871725744283416284, 8.693528026663551622312452601, 9.336617458773163321448436066870, 10.04368992064481324774078889713, 10.47280323219933279480441452502, 11.32475573558093910280780995343, 12.0394317050585402730752555072, 13.18218157742069152286338614812, 13.77115950505555296832585100628, 14.22150692720119077770253456126, 14.912911322611866930379918091088, 15.82577987716709511302093052468, 16.33918421891646628184963336453, 16.93533727749835579776956278937, 17.739108257385235946524879411870, 18.43909243969295603601894236544, 19.218078658360229194956365500222

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