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Properties

Label 1-373-373.343-r0-0-0
Degree $1$
Conductor $373$
Sign $0.992 - 0.121i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (−0.820 − 0.571i)2-s + (0.979 + 0.201i)3-s + (0.347 + 0.937i)4-s + (0.612 − 0.790i)5-s + (−0.688 − 0.724i)6-s + (0.528 + 0.848i)7-s + (0.250 − 0.968i)8-s + (0.918 + 0.394i)9-s + (−0.954 + 0.299i)10-s + (0.250 + 0.968i)11-s + (0.151 + 0.988i)12-s + (−0.250 − 0.968i)13-s + (0.0506 − 0.998i)14-s + (0.758 − 0.651i)15-s + (−0.758 + 0.651i)16-s + (0.347 + 0.937i)17-s + ⋯
L(s)  = 1  + (−0.820 − 0.571i)2-s + (0.979 + 0.201i)3-s + (0.347 + 0.937i)4-s + (0.612 − 0.790i)5-s + (−0.688 − 0.724i)6-s + (0.528 + 0.848i)7-s + (0.250 − 0.968i)8-s + (0.918 + 0.394i)9-s + (−0.954 + 0.299i)10-s + (0.250 + 0.968i)11-s + (0.151 + 0.988i)12-s + (−0.250 − 0.968i)13-s + (0.0506 − 0.998i)14-s + (0.758 − 0.651i)15-s + (−0.758 + 0.651i)16-s + (0.347 + 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.992 - 0.121i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.992 - 0.121i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.489506620 - 0.09103907247i\)
\(L(\frac12)\) \(\approx\) \(1.489506620 - 0.09103907247i\)
\(L(1)\) \(\approx\) \(1.184714197 - 0.1267317492i\)
\(L(1)\) \(\approx\) \(1.184714197 - 0.1267317492i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.820 - 0.571i)T \)
3 \( 1 + (0.979 + 0.201i)T \)
5 \( 1 + (0.612 - 0.790i)T \)
7 \( 1 + (0.528 + 0.848i)T \)
11 \( 1 + (0.250 + 0.968i)T \)
13 \( 1 + (-0.250 - 0.968i)T \)
17 \( 1 + (0.347 + 0.937i)T \)
19 \( 1 + (0.758 + 0.651i)T \)
23 \( 1 + (-0.979 + 0.201i)T \)
29 \( 1 + (-0.994 + 0.101i)T \)
31 \( 1 + (0.151 - 0.988i)T \)
37 \( 1 + (-0.874 - 0.485i)T \)
41 \( 1 + (0.688 + 0.724i)T \)
43 \( 1 + (-0.347 - 0.937i)T \)
47 \( 1 + (0.0506 + 0.998i)T \)
53 \( 1 + (-0.918 + 0.394i)T \)
59 \( 1 + (-0.994 + 0.101i)T \)
61 \( 1 + (-0.979 - 0.201i)T \)
67 \( 1 + (0.612 - 0.790i)T \)
71 \( 1 + (0.347 - 0.937i)T \)
73 \( 1 + (0.918 - 0.394i)T \)
79 \( 1 + (0.954 - 0.299i)T \)
83 \( 1 + (0.918 - 0.394i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.347 - 0.937i)T \)
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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

−24.528898050318559552880402726330, −24.31616365152566789073607132325, −23.16490533422942515693397400122, −21.87257478283946107258256682083, −20.91280749941461178386482136036, −20.05669276727789057699330247722, −19.181867823339969340334857856488, −18.4647729741155348766909151991, −17.74733743547202438403754861438, −16.69310197553288205723654459842, −15.81526093588681140986932419683, −14.61971865693416022479787646974, −13.96113824601513124749737357020, −13.723940248634750060404689253997, −11.68427969889596992684590032341, −10.76080677288778108537322697989, −9.76546751255446516679017619875, −9.10808133534054335376893376979, −7.97226895643045800175006380590, −7.15803564519902001650739926927, −6.49984725585648108140962772369, −5.06602082127048583807077964850, −3.535396169000519118513776415462, −2.291788550548293167328036505350, −1.2331826533736502134065221673, 1.59367421054401156154836829849, 2.10777578251838928385678161539, 3.42660015258350597175886669868, 4.615450988421863441611930027811, 5.90596181088645230492984620734, 7.72696076043809672848424610798, 8.08864705068055397595327912878, 9.281185147463407404875235194956, 9.664492609815468765209123413454, 10.67541453130054821850965720132, 12.2889410864509810465257492579, 12.5426954776897636626058753938, 13.758254785538430947149923023981, 14.93164080558311155906164670650, 15.69579890813698123798556094091, 16.82656243669327234075858921758, 17.73251812090105592447104277839, 18.41922732547565239126131945524, 19.48722817926442056103023711142, 20.383941498519523336743175503072, 20.71886795216869631903401945676, 21.65770210450905462019449684744, 22.380266673963863220990341063, 24.30681262677981397688777821608, 24.83293630942798420180370982067

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