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Properties

Label 1-4033-4033.2192-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.967 + 0.254i$
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 + (−0.993 + 0.116i)3-s + (0.766 + 0.642i)4-s + (0.973 + 0.230i)5-s + (−0.973 − 0.230i)6-s + (−0.286 − 0.957i)7-s + (0.5 + 0.866i)8-s + (0.973 − 0.230i)9-s + (0.835 + 0.549i)10-s + (0.835 − 0.549i)11-s + (−0.835 − 0.549i)12-s + (0.686 − 0.727i)13-s + (0.0581 − 0.998i)14-s + (−0.993 − 0.116i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.993 + 0.116i)3-s + (0.766 + 0.642i)4-s + (0.973 + 0.230i)5-s + (−0.973 − 0.230i)6-s + (−0.286 − 0.957i)7-s + (0.5 + 0.866i)8-s + (0.973 − 0.230i)9-s + (0.835 + 0.549i)10-s + (0.835 − 0.549i)11-s + (−0.835 − 0.549i)12-s + (0.686 − 0.727i)13-s + (0.0581 − 0.998i)14-s + (−0.993 − 0.116i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.967 + 0.254i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.967 + 0.254i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.413931082 + 0.4409918238i\)
\(L(\frac12)\) \(\approx\) \(3.413931082 + 0.4409918238i\)
\(L(1)\) \(\approx\) \(1.846712344 + 0.2884987461i\)
\(L(1)\) \(\approx\) \(1.846712344 + 0.2884987461i\)

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 \( 1 + (-0.993 + 0.116i)T \)
5 \( 1 + (0.973 + 0.230i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (0.835 - 0.549i)T \)
13 \( 1 + (0.686 - 0.727i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.835 - 0.549i)T \)
31 \( 1 + (-0.993 - 0.116i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.286 - 0.957i)T \)
53 \( 1 + (0.835 + 0.549i)T \)
59 \( 1 + (0.993 + 0.116i)T \)
61 \( 1 + (-0.0581 + 0.998i)T \)
67 \( 1 + (0.0581 + 0.998i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.893 + 0.448i)T \)
79 \( 1 + (-0.893 - 0.448i)T \)
83 \( 1 + (-0.286 - 0.957i)T \)
89 \( 1 + (0.396 - 0.918i)T \)
97 \( 1 + (0.396 - 0.918i)T \)
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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

−18.55097343940313322328571647910, −17.86440494024173322421162574164, −16.88162898357586663473749123214, −16.285356862997903852728250936454, −15.96556773888244355091732589445, −14.78210359136867113440884246500, −14.316267047831152841120594595286, −13.62243392657833286647259538187, −12.62165046556368672393920239002, −12.42647307341649240592993710294, −11.804343489571099183763524987948, −11.04441365253496182081436242707, −10.28776994544422661853551635413, −9.538573373171557373953705111073, −9.13697603600044413234100397559, −7.70134643737737033516196588829, −6.74296718194967747424364626089, −6.27633597818597207064246057685, −5.54831867681898354938651951263, −5.300548020132600092094983837011, −4.20334243240529649827933150725, −3.58185000648198581866220694467, −2.31400524985014929511344423670, −1.749460969654969972642060352269, −1.061116587593805700837251417938, 0.9217607929607025455350033534, 1.60127653401778945853051426759, 2.86557670039593391516955681539, 3.73847312742732037525774111321, 4.155993488050394628003496496513, 5.3336185034342360610339532103, 5.83963369717938277902476681965, 6.17613560143187591476966490637, 7.18036594127284672209289924805, 7.49984984468667851852389792444, 8.78746628916411081241211459044, 9.784384631418450669192267638, 10.344583067892844241224690976430, 11.145808505610210264001792251887, 11.50788848096398151838574572982, 12.5207968727749238967033732463, 13.21629615463491141049501752293, 13.53879229399657460481925275781, 14.36159344417308537808024656581, 14.98523925781850811291958484739, 16.01069580340751520151032606206, 16.393397035688315661440687379849, 17.08172369863627840001214801669, 17.53661988741856714413141564070, 18.172858599750116566705947849124

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