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Properties

Label 1-2672-2672.315-r0-0-0
Degree $1$
Conductor $2672$
Sign $-0.602 + 0.798i$
Analytic cond. $12.4087$
Root an. cond. $12.4087$
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  + (0.832 + 0.553i)3-s + (0.811 + 0.584i)5-s + (0.132 + 0.991i)7-s + (0.387 + 0.922i)9-s + (−0.978 − 0.206i)11-s + (0.0756 + 0.997i)13-s + (0.351 + 0.936i)15-s + (0.993 + 0.113i)17-s + (−0.995 − 0.0944i)19-s + (−0.438 + 0.898i)21-s + (0.843 + 0.537i)23-s + (0.316 + 0.948i)25-s + (−0.188 + 0.982i)27-s + (0.537 + 0.843i)29-s + (0.942 − 0.334i)31-s + ⋯
L(s)  = 1  + (0.832 + 0.553i)3-s + (0.811 + 0.584i)5-s + (0.132 + 0.991i)7-s + (0.387 + 0.922i)9-s + (−0.978 − 0.206i)11-s + (0.0756 + 0.997i)13-s + (0.351 + 0.936i)15-s + (0.993 + 0.113i)17-s + (−0.995 − 0.0944i)19-s + (−0.438 + 0.898i)21-s + (0.843 + 0.537i)23-s + (0.316 + 0.948i)25-s + (−0.188 + 0.982i)27-s + (0.537 + 0.843i)29-s + (0.942 − 0.334i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.602 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.602 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2672\)    =    \(2^{4} \cdot 167\)
Sign: $-0.602 + 0.798i$
Analytic conductor: \(12.4087\)
Root analytic conductor: \(12.4087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2672} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2672,\ (0:\ ),\ -0.602 + 0.798i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.215081819 + 2.438457061i\)
\(L(\frac12)\) \(\approx\) \(1.215081819 + 2.438457061i\)
\(L(1)\) \(\approx\) \(1.386453995 + 0.8516430199i\)
\(L(1)\) \(\approx\) \(1.386453995 + 0.8516430199i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 \)
good3 \( 1 + (0.832 + 0.553i)T \)
5 \( 1 + (0.811 + 0.584i)T \)
7 \( 1 + (0.132 + 0.991i)T \)
11 \( 1 + (-0.978 - 0.206i)T \)
13 \( 1 + (0.0756 + 0.997i)T \)
17 \( 1 + (0.993 + 0.113i)T \)
19 \( 1 + (-0.995 - 0.0944i)T \)
23 \( 1 + (0.843 + 0.537i)T \)
29 \( 1 + (0.537 + 0.843i)T \)
31 \( 1 + (0.942 - 0.334i)T \)
37 \( 1 + (0.922 + 0.387i)T \)
41 \( 1 + (0.776 - 0.629i)T \)
43 \( 1 + (-0.599 - 0.800i)T \)
47 \( 1 + (-0.726 - 0.686i)T \)
53 \( 1 + (0.505 - 0.862i)T \)
59 \( 1 + (-0.113 - 0.993i)T \)
61 \( 1 + (0.369 - 0.929i)T \)
67 \( 1 + (-0.811 + 0.584i)T \)
71 \( 1 + (0.169 + 0.985i)T \)
73 \( 1 + (0.822 + 0.569i)T \)
79 \( 1 + (-0.243 - 0.969i)T \)
83 \( 1 + (0.150 - 0.988i)T \)
89 \( 1 + (-0.351 + 0.936i)T \)
97 \( 1 + (-0.942 - 0.334i)T \)
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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

−19.249092045123627313756000264429, −18.06832702905055211284550539180, −17.94383218029335983256897717958, −16.95387876059663715379673700798, −16.41290612229139658975572525417, −15.32000403330562543326876510765, −14.7371231546245371495331154593, −13.91552846392082891024248560584, −13.351906372388017303770484847780, −12.820954870642178400836726322111, −12.31689439247474128148478199917, −10.98294423388812089070069499549, −10.16402202923042049197367126776, −9.82089865210203948137906421280, −8.76841313264499759302269492351, −7.98979384067790233584409029704, −7.68265682648863863995549612173, −6.57909733752889120417895821998, −5.912721186369291588819260138980, −4.86005726758379337982894663409, −4.21177297824530614903881451837, −2.95910148196204151560282390957, −2.54449952963223363039657244532, −1.326373702482570828760169742705, −0.7666702500462777453589779373, 1.55562689847771696643568712383, 2.341356551768738572669594170984, 2.85914908806423961708049700371, 3.7013378922054559811605996393, 4.88193135304905636542777499506, 5.387366784829406874066761784219, 6.32387002825945445286305772670, 7.147032821821203767635618036105, 8.17170363422582278131757044557, 8.68954364674430319159816661883, 9.51095301625099244151248412163, 10.042713097543722553446790144823, 10.81304937987745799468652569289, 11.51878112191017356634295332449, 12.62863339144068093642112404326, 13.28760380696223901535008804979, 13.987484044249047802747545698174, 14.70207112841488454119943641498, 15.110617385211411422349090422427, 15.92950990237405117188036341026, 16.62493722676210807195028670552, 17.47794885787524993802418795086, 18.37852704539008211058494502994, 18.99292631949088696629817519514, 19.205361923504995049253378687812

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