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Properties

Label 1-945-945.103-r0-0-0
Degree $1$
Conductor $945$
Sign $-0.997 - 0.0759i$
Analytic cond. $4.38856$
Root an. cond. $4.38856$
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  + (0.342 + 0.939i)2-s + (−0.766 + 0.642i)4-s + (−0.866 − 0.5i)8-s + (−0.939 + 0.342i)11-s + (0.984 − 0.173i)13-s + (0.173 − 0.984i)16-s + i·17-s + 19-s + (−0.642 − 0.766i)22-s + (−0.984 + 0.173i)23-s + (0.5 + 0.866i)26-s + (−0.173 + 0.984i)29-s + (−0.766 + 0.642i)31-s + (0.984 − 0.173i)32-s + (−0.939 + 0.342i)34-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (−0.766 + 0.642i)4-s + (−0.866 − 0.5i)8-s + (−0.939 + 0.342i)11-s + (0.984 − 0.173i)13-s + (0.173 − 0.984i)16-s + i·17-s + 19-s + (−0.642 − 0.766i)22-s + (−0.984 + 0.173i)23-s + (0.5 + 0.866i)26-s + (−0.173 + 0.984i)29-s + (−0.766 + 0.642i)31-s + (0.984 − 0.173i)32-s + (−0.939 + 0.342i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.997 - 0.0759i$
Analytic conductor: \(4.38856\)
Root analytic conductor: \(4.38856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 945,\ (0:\ ),\ -0.997 - 0.0759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03835053037 + 1.008943551i\)
\(L(\frac12)\) \(\approx\) \(-0.03835053037 + 1.008943551i\)
\(L(1)\) \(\approx\) \(0.7558993506 + 0.6426977726i\)
\(L(1)\) \(\approx\) \(0.7558993506 + 0.6426977726i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.342 + 0.939i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + iT \)
19 \( 1 + T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (-0.173 + 0.984i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.642 + 0.766i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.984 - 0.173i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.642 + 0.766i)T \)
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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

−21.36057974310384869842138661669, −20.50124331755464567925474023691, −20.184819663304476764081008748482, −19.00447504543184077553789344461, −18.29738498672108757247745920784, −17.98889699246371952838673000997, −16.54368966770347072596258549983, −15.82358385256501717059417672904, −14.92773742018358137805762021454, −13.772434402590587029038212997554, −13.55125344950367008185041776941, −12.57241341950775785822079165525, −11.55765326250390692045719161569, −11.15191302045959699965013920818, −10.08209878258818549983865403403, −9.46260892753116068174457608161, −8.43969233436842824910803027842, −7.58682449400271899468299777516, −6.168860726730397094542177741783, −5.4747261559499985232410195225, −4.52019949548476499657308454700, −3.53393902925840869756394182208, −2.71906434140359665890032669262, −1.68164859486157478653377365249, −0.39517692409086907813773438997, 1.48200237298656419470379692540, 3.0233448658774513117409561717, 3.80704885382724892109095466643, 4.87915246788786913992333101253, 5.67607330442243805979605672394, 6.42800252942599052561467023064, 7.50819463541374465321540910101, 8.103197305588851029259511030, 8.965462162611077063279646040493, 9.96196684633117197247991844693, 10.87053635854790130113129111369, 12.01520342086853181075254212980, 12.88610662599022212165577445779, 13.43607845718533981351226421496, 14.36563327860087821266820959366, 15.09976549045509314839493648561, 16.02058246531895064083390226277, 16.306785584756662827451816000836, 17.57216829191504921046865744419, 18.0696462505623588981371310710, 18.73675803081602857887336555446, 19.98858629892980932803502415924, 20.76834273099561064471754544263, 21.63113997063577160304272177844, 22.281710213370601247867696478016

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