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Properties

Label 1-4033-4033.156-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.414 + 0.909i$
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  + (0.939 + 0.342i)2-s + (−0.597 + 0.802i)3-s + (0.766 + 0.642i)4-s + (0.727 − 0.686i)5-s + (−0.835 + 0.549i)6-s + (0.973 − 0.230i)7-s + (0.5 + 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.973 + 0.230i)12-s + (−0.973 + 0.230i)13-s + (0.993 + 0.116i)14-s + (0.116 + 0.993i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.597 + 0.802i)3-s + (0.766 + 0.642i)4-s + (0.727 − 0.686i)5-s + (−0.835 + 0.549i)6-s + (0.973 − 0.230i)7-s + (0.5 + 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.973 + 0.230i)12-s + (−0.973 + 0.230i)13-s + (0.993 + 0.116i)14-s + (0.116 + 0.993i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.414 + 0.909i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.414 + 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.710789090 + 1.742996897i\)
\(L(\frac12)\) \(\approx\) \(2.710789090 + 1.742996897i\)
\(L(1)\) \(\approx\) \(1.706349713 + 0.6611458318i\)
\(L(1)\) \(\approx\) \(1.706349713 + 0.6611458318i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (0.727 - 0.686i)T \)
7 \( 1 + (0.973 - 0.230i)T \)
11 \( 1 + (-0.918 - 0.396i)T \)
13 \( 1 + (-0.973 + 0.230i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.802 + 0.597i)T \)
31 \( 1 + (0.957 - 0.286i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.448 - 0.893i)T \)
53 \( 1 + (0.957 + 0.286i)T \)
59 \( 1 + (0.993 - 0.116i)T \)
61 \( 1 + (0.998 - 0.0581i)T \)
67 \( 1 + (0.727 + 0.686i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.396 + 0.918i)T \)
79 \( 1 + (-0.973 - 0.230i)T \)
83 \( 1 + (-0.597 + 0.802i)T \)
89 \( 1 + (-0.998 + 0.0581i)T \)
97 \( 1 + (0.727 - 0.686i)T \)
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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

−18.37524619512204066626584757148, −17.73423260821901554538901513761, −17.23134397745275748354596806748, −16.18532493002802981692124555944, −15.519423550811370695761223878699, −14.567090498887213298220352085905, −14.17763741126870168061465199292, −13.66117112558807156888480754047, −12.77247844804276959208682926203, −12.19975735595057400063912094151, −11.675459460984220694824728996617, −10.91873942477478369199501479017, −10.19739344681802645242914682128, −9.86808218653666948681988115421, −8.29916424707245423392301943460, −7.55774553469461731977498262314, −7.061068212372297231808108140869, −6.163539183039504182425957672878, −5.55123092490706991802441105279, −5.04002155603423685122122773069, −4.31137926226363052505981608764, −2.89067040313263985362795120955, −2.33027503270494151657857015426, −1.91942177063937784932606547402, −0.79589411299084394994677062036, 0.92283062669370081703538296240, 2.09493750397919123521700190523, 2.76141680892769488603989187768, 3.98283655135204787616674471830, 4.52873917639725856913603384195, 5.16628390989848171762965381854, 5.56270309701783465826962157818, 6.36168153867717023371129818560, 7.21376620388918500148004728101, 8.2771084529249349652682976882, 8.6025499350324771168996580170, 9.91361923879594409159145500610, 10.30717481316005650021313407171, 11.19479126238311652681001654049, 11.740045582218988873595279610274, 12.51177291367897530206249690269, 13.11821286613720368484418489614, 13.96380854309014250598495992750, 14.469614547698682688834176711590, 15.23059908188920963603296891578, 15.83558822961690198035928357026, 16.50545248952314941932870339931, 17.207166057950315612940899844401, 17.49395055527697425510485321712, 18.1918042840614136340055841721

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