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Properties

Label 1-4033-4033.1390-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.949 - 0.312i$
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 + 3-s + (0.766 − 0.642i)4-s + (0.5 − 0.866i)5-s + (−0.939 + 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + 9-s + (−0.173 + 0.984i)10-s + (−0.173 − 0.984i)11-s + (0.766 − 0.642i)12-s + 13-s + (0.173 − 0.984i)14-s + (0.5 − 0.866i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + 3-s + (0.766 − 0.642i)4-s + (0.5 − 0.866i)5-s + (−0.939 + 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + 9-s + (−0.173 + 0.984i)10-s + (−0.173 − 0.984i)11-s + (0.766 − 0.642i)12-s + 13-s + (0.173 − 0.984i)14-s + (0.5 − 0.866i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.949 - 0.312i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1390, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.949 - 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.984124310 - 0.3178042608i\)
\(L(\frac12)\) \(\approx\) \(1.984124310 - 0.3178042608i\)
\(L(1)\) \(\approx\) \(1.168519468 + 0.01033598601i\)
\(L(1)\) \(\approx\) \(1.168519468 + 0.01033598601i\)

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 \( 1 + T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.173 - 0.984i)T \)
13 \( 1 + T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−18.43533512310365466515693962063, −18.05046564506338829968340758344, −17.41635138111388742275400351325, −16.40095236173914740014279563693, −15.85365119644216887847961425746, −15.25124865900362974342459570860, −14.20892364276806304489805109862, −13.93913023980070306153858288264, −12.92179871334806136020135251394, −12.52711548780024627842924091999, −11.326984224633294329130305270595, −10.64008707718123898936899111162, −10.098385865452823082001594956957, −9.58708418250170506367593577271, −8.9729987677640162864155243361, −7.96144082421486973242153685431, −7.498756716225277752854916468983, −6.68256042907855066872639384390, −6.41910466423435003754792033142, −4.79356416281680337181757296740, −3.89106198898052123003991048841, −3.00390478849649808657337703245, −2.779320581579992165880925121643, −1.69283771093804873233117894499, −0.99152941671619499867992612763, 0.77560972320956958251288197727, 1.63484151088057110446796903311, 2.23692367964507996857069816730, 3.25498420478434707342432616879, 3.90238452250941414302219478549, 5.35431752215760779431666190647, 5.855217436496079726901555072943, 6.38136053815068572579272620551, 7.64463143881388933473290614568, 8.17193223398679548182671315156, 8.80367823649789537074979440785, 9.08166995239480107955928383127, 10.103137693678822213731283198364, 10.29439739176294954395272123366, 11.65055476582800953999200550091, 12.19956116063267272729174295021, 13.145122392679938715884453232191, 13.702128840245404216223634709376, 14.40391276060872500007355146330, 15.25949147079175747146565841728, 15.88483230662614120445692695751, 16.28993025402598813878134286274, 16.944404244689687348153393180947, 17.97507670877018555164432667900, 18.441806835059363710395339533284

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