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Properties

Label 1-373-373.12-r0-0-0
Degree $1$
Conductor $373$
Sign $0.920 - 0.390i$
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  + (−0.612 − 0.790i)2-s + (−0.954 − 0.299i)3-s + (−0.250 + 0.968i)4-s + (0.979 + 0.201i)5-s + (0.347 + 0.937i)6-s + (−0.0506 − 0.998i)7-s + (0.918 − 0.394i)8-s + (0.820 + 0.571i)9-s + (−0.440 − 0.897i)10-s + (0.918 + 0.394i)11-s + (0.528 − 0.848i)12-s + (0.918 + 0.394i)13-s + (−0.758 + 0.651i)14-s + (−0.874 − 0.485i)15-s + (−0.874 − 0.485i)16-s + (−0.250 + 0.968i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.790i)2-s + (−0.954 − 0.299i)3-s + (−0.250 + 0.968i)4-s + (0.979 + 0.201i)5-s + (0.347 + 0.937i)6-s + (−0.0506 − 0.998i)7-s + (0.918 − 0.394i)8-s + (0.820 + 0.571i)9-s + (−0.440 − 0.897i)10-s + (0.918 + 0.394i)11-s + (0.528 − 0.848i)12-s + (0.918 + 0.394i)13-s + (−0.758 + 0.651i)14-s + (−0.874 − 0.485i)15-s + (−0.874 − 0.485i)16-s + (−0.250 + 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.920 - 0.390i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8257862450 - 0.1681195964i\)
\(L(\frac12)\) \(\approx\) \(0.8257862450 - 0.1681195964i\)
\(L(1)\) \(\approx\) \(0.7180032670 - 0.2082180859i\)
\(L(1)\) \(\approx\) \(0.7180032670 - 0.2082180859i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.612 - 0.790i)T \)
3 \( 1 + (-0.954 - 0.299i)T \)
5 \( 1 + (0.979 + 0.201i)T \)
7 \( 1 + (-0.0506 - 0.998i)T \)
11 \( 1 + (0.918 + 0.394i)T \)
13 \( 1 + (0.918 + 0.394i)T \)
17 \( 1 + (-0.250 + 0.968i)T \)
19 \( 1 + (-0.874 + 0.485i)T \)
23 \( 1 + (-0.954 + 0.299i)T \)
29 \( 1 + (0.151 + 0.988i)T \)
31 \( 1 + (0.528 + 0.848i)T \)
37 \( 1 + (0.688 - 0.724i)T \)
41 \( 1 + (0.347 + 0.937i)T \)
43 \( 1 + (-0.250 + 0.968i)T \)
47 \( 1 + (-0.758 - 0.651i)T \)
53 \( 1 + (0.820 - 0.571i)T \)
59 \( 1 + (0.151 + 0.988i)T \)
61 \( 1 + (-0.954 - 0.299i)T \)
67 \( 1 + (0.979 + 0.201i)T \)
71 \( 1 + (-0.250 - 0.968i)T \)
73 \( 1 + (0.820 - 0.571i)T \)
79 \( 1 + (-0.440 - 0.897i)T \)
83 \( 1 + (0.820 - 0.571i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.250 + 0.968i)T \)
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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

−24.68998493112881351121390556818, −24.09384169976645634200792639267, −22.83150336468874135666165516381, −22.25133874276838449052598204297, −21.38098243124311429816335201399, −20.33250727575423984127213309753, −18.89363147209017992157642732223, −18.27783968736069510492697894558, −17.48977491006612488716161544999, −16.83621988625257872163655102309, −15.905605455146645516258828755738, −15.24894172208621135419195507370, −14.058322331361264515131840906025, −13.122635468397064551052528126805, −11.83288518643282025560613102804, −10.938062963143228431960640115840, −9.85276896594357069367328663162, −9.1979382568126936718919079610, −8.31194611996838161160594245701, −6.59858080077233181057594734190, −6.13761930516025301730837215186, −5.4009110525993068574124512863, −4.29521563023306218104630378462, −2.21026188108041829512992719470, −0.84030255336001676509065849732, 1.27812111708160135235444502353, 1.840443599286395144022886701477, 3.684583850422536983096342468941, 4.568226010091663940246092215297, 6.23481002425878542485028592694, 6.76463115451572485663588879455, 8.05697283867652175937674070482, 9.292374929412566382288173737784, 10.318989496436500459553010567567, 10.730837916704973682723023839018, 11.7527195351416318996349637216, 12.803010536563859908100573790825, 13.43635195367693706209860719881, 14.46582838377043633169150536516, 16.358170452330974301950576708750, 16.79149749225665866705947666229, 17.74332117118640401666786604959, 18.07225637350851656917544257409, 19.302435423943375376732480248442, 20.022768122608118165235021366945, 21.31833931802082271042810676517, 21.656053649846062090740033169710, 22.775687137021965189729500181247, 23.381149090414056155350713079926, 24.65466599692831372732136891443

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