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Properties

Label 1-373-373.97-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.538 + 0.842i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (0.848 + 0.528i)2-s + (0.758 + 0.651i)3-s + (0.440 + 0.897i)4-s + (0.998 − 0.0506i)5-s + (0.299 + 0.954i)6-s + (−0.918 − 0.394i)7-s + (−0.101 + 0.994i)8-s + (0.151 + 0.988i)9-s + (0.874 + 0.485i)10-s + (−0.101 − 0.994i)11-s + (−0.250 + 0.968i)12-s + (0.994 − 0.101i)13-s + (−0.571 − 0.820i)14-s + (0.790 + 0.612i)15-s + (−0.612 + 0.790i)16-s + (0.440 + 0.897i)17-s + ⋯
L(s)  = 1  + (0.848 + 0.528i)2-s + (0.758 + 0.651i)3-s + (0.440 + 0.897i)4-s + (0.998 − 0.0506i)5-s + (0.299 + 0.954i)6-s + (−0.918 − 0.394i)7-s + (−0.101 + 0.994i)8-s + (0.151 + 0.988i)9-s + (0.874 + 0.485i)10-s + (−0.101 − 0.994i)11-s + (−0.250 + 0.968i)12-s + (0.994 − 0.101i)13-s + (−0.571 − 0.820i)14-s + (0.790 + 0.612i)15-s + (−0.612 + 0.790i)16-s + (0.440 + 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.538 + 0.842i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.538 + 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.340377895 + 4.272153422i\)
\(L(\frac12)\) \(\approx\) \(2.340377895 + 4.272153422i\)
\(L(1)\) \(\approx\) \(1.951810485 + 1.470641768i\)
\(L(1)\) \(\approx\) \(1.951810485 + 1.470641768i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.848 + 0.528i)T \)
3 \( 1 + (0.758 + 0.651i)T \)
5 \( 1 + (0.998 - 0.0506i)T \)
7 \( 1 + (-0.918 - 0.394i)T \)
11 \( 1 + (-0.101 - 0.994i)T \)
13 \( 1 + (0.994 - 0.101i)T \)
17 \( 1 + (0.440 + 0.897i)T \)
19 \( 1 + (-0.790 + 0.612i)T \)
23 \( 1 + (0.651 + 0.758i)T \)
29 \( 1 + (0.347 + 0.937i)T \)
31 \( 1 + (0.250 + 0.968i)T \)
37 \( 1 + (-0.979 - 0.201i)T \)
41 \( 1 + (-0.954 + 0.299i)T \)
43 \( 1 + (-0.897 + 0.440i)T \)
47 \( 1 + (0.571 - 0.820i)T \)
53 \( 1 + (0.988 + 0.151i)T \)
59 \( 1 + (-0.347 - 0.937i)T \)
61 \( 1 + (0.651 - 0.758i)T \)
67 \( 1 + (0.998 - 0.0506i)T \)
71 \( 1 + (0.440 - 0.897i)T \)
73 \( 1 + (0.151 - 0.988i)T \)
79 \( 1 + (0.485 - 0.874i)T \)
83 \( 1 + (0.151 - 0.988i)T \)
89 \( 1 - T \)
97 \( 1 + (0.897 - 0.440i)T \)
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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

−24.09188883039394395572483460055, −23.02095395272851391422886703710, −22.49637075636619916484388112949, −21.21545601916596049966992562001, −20.75454412019883280579229660069, −19.894947128977169074502255520578, −18.7815796049326424941603850023, −18.43805046368044741717291965203, −17.10116597153627728836930227017, −15.660608106096008535094691454755, −14.992499325033040042785886046, −13.91708085899423475271511788549, −13.30120048334328526730046488389, −12.700248296232824707211886578500, −11.77164325851561196098858995991, −10.317034482052113556406290181766, −9.59155213126415454823639054532, −8.72421670326093809516657841291, −6.93831980523565031751271186012, −6.46835453501745123872293836804, −5.35661001560417075571400666151, −3.97963023761030941531651957544, −2.73562474866748474188570688948, −2.2220838445459590472734041314, −0.90342886129913350305857343673, 1.73058925117586998396403831909, 3.287250552551950693554905413219, 3.53079664165576763575591162780, 5.05351972300195857845942739428, 5.96942830518407696707301429238, 6.81486471011521882348648314676, 8.30225740643282943664504937338, 8.887842573140741601004612326898, 10.23599597196496216402222772682, 10.86496824725710470158647491027, 12.5746756905829493047469839480, 13.4122869816388435911321688355, 13.839730209960709588571471239598, 14.78046191423548879089034436956, 15.78293147458973630319448176919, 16.52277958987244848542163269832, 17.14169641229895262644029106883, 18.62522382038467203692652523259, 19.6578230828960719275083203339, 20.65713013525932562232659378832, 21.4969328282435049982000956497, 21.74860912584567379018158610276, 22.96568280433683977863787761954, 23.72142487065311439466911289787, 25.06522659476399524804850823051

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