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Properties

Label 1-3381-3381.212-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.981 + 0.193i$
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  + (0.984 − 0.175i)2-s + (0.938 − 0.346i)4-s + (−0.612 + 0.790i)5-s + (0.862 − 0.505i)8-s + (−0.464 + 0.885i)10-s + (0.963 − 0.268i)11-s + (−0.591 + 0.806i)13-s + (0.760 − 0.649i)16-s + (−0.169 − 0.985i)17-s + (−0.580 + 0.814i)19-s + (−0.301 + 0.953i)20-s + (0.900 − 0.433i)22-s + (−0.248 − 0.968i)25-s + (−0.440 + 0.897i)26-s + (0.768 − 0.639i)29-s + ⋯
L(s)  = 1  + (0.984 − 0.175i)2-s + (0.938 − 0.346i)4-s + (−0.612 + 0.790i)5-s + (0.862 − 0.505i)8-s + (−0.464 + 0.885i)10-s + (0.963 − 0.268i)11-s + (−0.591 + 0.806i)13-s + (0.760 − 0.649i)16-s + (−0.169 − 0.985i)17-s + (−0.580 + 0.814i)19-s + (−0.301 + 0.953i)20-s + (0.900 − 0.433i)22-s + (−0.248 − 0.968i)25-s + (−0.440 + 0.897i)26-s + (0.768 − 0.639i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.981 + 0.193i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.981 + 0.193i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.199004950 + 0.3131929917i\)
\(L(\frac12)\) \(\approx\) \(3.199004950 + 0.3131929917i\)
\(L(1)\) \(\approx\) \(1.874260784 + 0.02082466407i\)
\(L(1)\) \(\approx\) \(1.874260784 + 0.02082466407i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.984 - 0.175i)T \)
5 \( 1 + (-0.612 + 0.790i)T \)
11 \( 1 + (0.963 - 0.268i)T \)
13 \( 1 + (-0.591 + 0.806i)T \)
17 \( 1 + (-0.169 - 0.985i)T \)
19 \( 1 + (-0.580 + 0.814i)T \)
29 \( 1 + (0.768 - 0.639i)T \)
31 \( 1 + (-0.786 - 0.618i)T \)
37 \( 1 + (0.352 + 0.935i)T \)
41 \( 1 + (0.377 + 0.925i)T \)
43 \( 1 + (0.862 + 0.505i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (0.704 + 0.709i)T \)
59 \( 1 + (-0.464 + 0.885i)T \)
61 \( 1 + (0.997 + 0.0679i)T \)
67 \( 1 + (0.888 - 0.458i)T \)
71 \( 1 + (0.992 + 0.122i)T \)
73 \( 1 + (0.314 - 0.949i)T \)
79 \( 1 + (0.327 - 0.945i)T \)
83 \( 1 + (-0.0611 - 0.998i)T \)
89 \( 1 + (-0.427 - 0.903i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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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

−19.301754621084760684048614287395, −17.76538066429620653194252467551, −17.24082657112083931348917045815, −16.68746239086034025349931081756, −15.78231067796901682172124414612, −15.39605150797981117447889927810, −14.60331250982365706271756307310, −14.083545532974430194783388237581, −12.98915834043248839480794256737, −12.526919841171737651075154709237, −12.22531910392113998135342506537, −11.13361374646510798622079562368, −10.751115346445602560030159755758, −9.58304491905944297319595046522, −8.69899288199132540688856351240, −8.11029805965560097356658479759, −7.17970590955820101387286285153, −6.69557624310828137293776056136, −5.59099446290741512176656744822, −5.12184295700203180269753260551, −4.071698519246426007582163448795, −3.90587833927368948759417028857, −2.72884573032078807932196978617, −1.85701495744166129085319159206, −0.815751811449847394446427008671, 0.92304978397883727066991104981, 2.12703606352901334153546015693, 2.714630384697398578373441493210, 3.68584900185247604557224062786, 4.19268846291082567225139395727, 4.90525021274980360894878654928, 6.109024126205012398373688131665, 6.47989765196645918111515319238, 7.29007741529889217230787870396, 7.87599278630448978887540049134, 9.0531516415340165975214691913, 9.87425353481454722215818378426, 10.61950635100090409059199725008, 11.495792847261225642361310829512, 11.750161187065367051844148962595, 12.418297653721807655899230216947, 13.43954572956362718285553458798, 14.10838251095415176829459952212, 14.600985116088535301085465055103, 15.127937794854341019636663863298, 15.99362616118610292156270545305, 16.572999368418431564900111039323, 17.26592289891809715626805952605, 18.53624243551938599215219336682, 18.86320886491960247684888518515

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