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Properties

Label 1-4033-4033.293-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.400 + 0.916i$
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  + (0.766 − 0.642i)2-s + 3-s + (0.173 − 0.984i)4-s + (−0.5 − 0.866i)5-s + (0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + 9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + (0.173 − 0.984i)12-s + 13-s + (−0.939 − 0.342i)14-s + (−0.5 − 0.866i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + 3-s + (0.173 − 0.984i)4-s + (−0.5 − 0.866i)5-s + (0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + 9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + (0.173 − 0.984i)12-s + 13-s + (−0.939 − 0.342i)14-s + (−0.5 − 0.866i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.400 + 0.916i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.400 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7057429575 - 1.078779440i\)
\(L(\frac12)\) \(\approx\) \(-0.7057429575 - 1.078779440i\)
\(L(1)\) \(\approx\) \(1.103674577 - 1.041680630i\)
\(L(1)\) \(\approx\) \(1.103674577 - 1.041680630i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.173 - 0.984i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−18.75823337558999054493883946239, −18.36478229629537095075719039568, −17.81651746651604136675420640588, −16.27739366507720729940830943607, −16.08226927044669217080769822958, −15.29595506940761032338817562159, −15.082069042765461643938735581372, −14.150499121775679389547386333853, −13.620183156109693606277152353542, −13.01259769666497175416261170191, −12.330045732350822889832327450878, −11.468895976007523796526251304701, −10.81830364456646906185940189025, −9.82728578645807220455379449148, −8.971138187756308354690734214468, −8.275321826539695094844592975880, −7.83201989874593242027662669782, −6.92895053990537960408171576896, −6.436417095613468912619477619247, −5.55483604018339329893933118696, −4.72052612200480870729173899228, −3.72419863905056910058799420096, −3.18815727517446426109416553698, −2.7253307173276745736517341462, −1.87869386104765494253138825542, 0.20566864888742257333912514343, 1.41755504309802369519652530152, 1.92837026478509850528524033841, 3.10219012629423263084088638952, 3.63661101938454668114904527354, 4.320994106485954866425768861100, 4.77394778150510198335660688752, 5.9766372788454250414409840691, 6.64263374001461061947005274993, 7.65544797756360938913535074730, 8.30479011140082441417270102491, 8.92620652528173594598621082636, 9.939239463693015657244729388932, 10.38715137421064296397747101665, 10.991750529486652805059503241894, 12.204935766242253109979221570137, 12.68568732502106442195760964903, 13.218524603055832812328784955821, 13.64354844868518940932476704445, 14.53429089817107051050788351895, 15.16055777792196748433341408270, 15.88456401360832531205164225442, 16.271797854197113019735364429435, 17.26189926064371472808915608936, 18.429617462927713573519269377912

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