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Properties

Label 1-671-671.194-r0-0-0
Degree $1$
Conductor $671$
Sign $0.732 + 0.680i$
Analytic cond. $3.11611$
Root an. cond. $3.11611$
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  + (0.951 − 0.309i)2-s + (0.809 + 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.951 + 0.309i)6-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (0.309 + 0.951i)9-s + i·10-s + 12-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (0.951 + 0.309i)17-s + (0.587 + 0.809i)18-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 + 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.951 + 0.309i)6-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (0.309 + 0.951i)9-s + i·10-s + 12-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (0.951 + 0.309i)17-s + (0.587 + 0.809i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.732 + 0.680i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.732 + 0.680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.217047738 + 1.264596437i\)
\(L(\frac12)\) \(\approx\) \(3.217047738 + 1.264596437i\)
\(L(1)\) \(\approx\) \(2.323063725 + 0.4576635280i\)
\(L(1)\) \(\approx\) \(2.323063725 + 0.4576635280i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.587 - 0.809i)T \)
31 \( 1 + (0.951 - 0.309i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.587 + 0.809i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.951 - 0.309i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.951 - 0.309i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−23.10260348904479002064266870780, −21.66845857478505830975313654759, −20.770344974255298714860942992890, −20.58985603376855113237382926183, −19.6263496985840342700159450649, −18.83000849324084811968376448861, −17.49868632917000751007421637880, −16.704567988913368316207129114758, −16.11065668431254527950113228216, −14.79955775740231063211355042588, −14.3802971313256223624273661434, −13.60012854511108366657996222247, −12.723444443389357907938617220260, −12.168997752227840077736673769281, −11.26095697398983677093494085678, −9.937809921276177815591081703601, −8.65880979076765868508749099567, −8.02783285630051996093040035908, −7.25362905524297003225814663524, −6.39150615413713601776858942343, −5.05367490870299644020273844823, −4.26511763638511161526454143471, −3.5170524466898824699998359716, −2.1537769518133412429812940151, −1.25036081035127491495740679312, 1.81564614479548786005523706873, 2.75420232303522413877101173152, 3.33309693657713475240021651516, 4.4006820794602534825416390899, 5.32154450508157860892062704880, 6.2670785042756919304683300542, 7.58709978756215162193438466729, 8.15051472700634757060244349090, 9.60852299024232392204478020991, 10.31451091837757929382556147322, 11.17538229148679270550997164390, 11.91000925804719643884752108561, 13.023465265333519249241753355944, 13.85857438010386561565225527791, 14.79198459901714375317889480664, 15.15539209502478937802440764500, 15.6191657062654516989649731683, 16.95537683337756016663916052703, 18.22903834931396826161849451893, 19.197594576206507466125585067739, 19.56137329143056133754950343567, 20.70295580576119943469728296644, 21.30028701653082587912680536906, 21.98168869202677106707910713819, 22.605203222580888240461493008305

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