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Properties

Label 1-3381-3381.65-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.327 + 0.944i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (0.0339 + 0.999i)2-s + (−0.997 + 0.0679i)4-s + (0.998 − 0.0543i)5-s + (−0.101 − 0.994i)8-s + (0.0882 + 0.996i)10-s + (−0.403 − 0.915i)11-s + (−0.0611 + 0.998i)13-s + (0.990 − 0.135i)16-s + (0.440 + 0.897i)17-s + (0.786 − 0.618i)19-s + (−0.992 + 0.122i)20-s + (0.900 − 0.433i)22-s + (0.994 − 0.108i)25-s + (−0.999 − 0.0271i)26-s + (−0.557 + 0.830i)29-s + ⋯
L(s)  = 1  + (0.0339 + 0.999i)2-s + (−0.997 + 0.0679i)4-s + (0.998 − 0.0543i)5-s + (−0.101 − 0.994i)8-s + (0.0882 + 0.996i)10-s + (−0.403 − 0.915i)11-s + (−0.0611 + 0.998i)13-s + (0.990 − 0.135i)16-s + (0.440 + 0.897i)17-s + (0.786 − 0.618i)19-s + (−0.992 + 0.122i)20-s + (0.900 − 0.433i)22-s + (0.994 − 0.108i)25-s + (−0.999 − 0.0271i)26-s + (−0.557 + 0.830i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.327 + 0.944i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.327 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.026164577 + 1.442044878i\)
\(L(\frac12)\) \(\approx\) \(1.026164577 + 1.442044878i\)
\(L(1)\) \(\approx\) \(0.9924065355 + 0.5938630875i\)
\(L(1)\) \(\approx\) \(0.9924065355 + 0.5938630875i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.0339 + 0.999i)T \)
5 \( 1 + (0.998 - 0.0543i)T \)
11 \( 1 + (-0.403 - 0.915i)T \)
13 \( 1 + (-0.0611 + 0.998i)T \)
17 \( 1 + (0.440 + 0.897i)T \)
19 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (-0.557 + 0.830i)T \)
31 \( 1 + (-0.888 + 0.458i)T \)
37 \( 1 + (-0.938 - 0.346i)T \)
41 \( 1 + (0.452 - 0.891i)T \)
43 \( 1 + (-0.101 + 0.994i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (0.209 + 0.977i)T \)
59 \( 1 + (0.0882 + 0.996i)T \)
61 \( 1 + (0.476 - 0.879i)T \)
67 \( 1 + (0.327 + 0.945i)T \)
71 \( 1 + (-0.262 + 0.965i)T \)
73 \( 1 + (-0.568 + 0.822i)T \)
79 \( 1 + (-0.235 - 0.971i)T \)
83 \( 1 + (0.794 + 0.607i)T \)
89 \( 1 + (0.644 - 0.764i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−18.41422621231354918758672232122, −18.15814700914930499611696512404, −17.44451804128177778741436524566, −16.841755251326592234990402084977, −15.82139225118013555228923737523, −14.886212365730049178982216300549, −14.330694123229489504860255652301, −13.51773465721815598554550756384, −13.06167165383617126925655150724, −12.330088509282692741360524186643, −11.70560364440505497590637586919, −10.72034190185492834371361110684, −10.1741246963036574289845757373, −9.62051521710529062015909115533, −9.09391234690688502298187504527, −7.947842754101478124543478342892, −7.41866572237879554218713513108, −6.19183641274683355555434746388, −5.2614157765492424298795744586, −5.09906809571574781874910582709, −3.85585765513576595460204497562, −3.05422113517062592931737879115, −2.3171326118786815165680017346, −1.638351798088032709459809630840, −0.60887762024374573442508870285, 0.95262767448994098094416188759, 1.87462462841830295998948501722, 3.0544390129149926189172744108, 3.84150571971917692597280605318, 4.83871225353029233014485679192, 5.615158690784292895151200599507, 5.92844898431836451352282255295, 6.96636474553017027672664760, 7.3894671109100775014744564316, 8.68319865672819817928067624014, 8.82742373278265898100685229681, 9.71199943896468995856917292110, 10.440601792742811250847311744699, 11.21733962219757115428035251751, 12.35619170877118644729144831473, 12.99172907237780836073972238935, 13.695245549651954446054432792682, 14.19750694494086954439298275614, 14.72760984778828625417142988223, 15.75178774514873772724181402927, 16.324638136883810887322269671417, 16.88459930180469164044297590593, 17.51799007394362568883376918254, 18.238309633222422778385469985164, 18.77867632887813300288848580580

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