Properties

Label 1-3381-3381.1124-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.986 - 0.163i$
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  + (−0.894 − 0.446i)2-s + (0.601 + 0.798i)4-s + (0.869 − 0.494i)5-s + (−0.182 − 0.983i)8-s + (−0.998 + 0.0543i)10-s + (−0.999 + 0.0135i)11-s + (−0.979 + 0.202i)13-s + (−0.275 + 0.961i)16-s + (0.390 + 0.920i)17-s + (−0.0475 − 0.998i)19-s + (0.917 + 0.396i)20-s + (0.900 + 0.433i)22-s + (0.511 − 0.859i)25-s + (0.966 + 0.255i)26-s + (0.992 + 0.122i)29-s + ⋯
L(s)  = 1  + (−0.894 − 0.446i)2-s + (0.601 + 0.798i)4-s + (0.869 − 0.494i)5-s + (−0.182 − 0.983i)8-s + (−0.998 + 0.0543i)10-s + (−0.999 + 0.0135i)11-s + (−0.979 + 0.202i)13-s + (−0.275 + 0.961i)16-s + (0.390 + 0.920i)17-s + (−0.0475 − 0.998i)19-s + (0.917 + 0.396i)20-s + (0.900 + 0.433i)22-s + (0.511 − 0.859i)25-s + (0.966 + 0.255i)26-s + (0.992 + 0.122i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.986 - 0.163i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.986 - 0.163i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.089518065 - 0.08957713180i\)
\(L(\frac12)\) \(\approx\) \(1.089518065 - 0.08957713180i\)
\(L(1)\) \(\approx\) \(0.7689810555 - 0.1422686349i\)
\(L(1)\) \(\approx\) \(0.7689810555 - 0.1422686349i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.894 - 0.446i)T \)
5 \( 1 + (0.869 - 0.494i)T \)
11 \( 1 + (-0.999 + 0.0135i)T \)
13 \( 1 + (-0.979 + 0.202i)T \)
17 \( 1 + (0.390 + 0.920i)T \)
19 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (0.992 + 0.122i)T \)
31 \( 1 + (0.981 + 0.189i)T \)
37 \( 1 + (-0.976 - 0.215i)T \)
41 \( 1 + (0.862 + 0.505i)T \)
43 \( 1 + (-0.182 + 0.983i)T \)
47 \( 1 + (0.733 + 0.680i)T \)
53 \( 1 + (0.938 + 0.346i)T \)
59 \( 1 + (-0.998 + 0.0543i)T \)
61 \( 1 + (-0.704 - 0.709i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
71 \( 1 + (-0.986 - 0.162i)T \)
73 \( 1 + (-0.248 + 0.968i)T \)
79 \( 1 + (0.995 + 0.0950i)T \)
83 \( 1 + (0.996 - 0.0815i)T \)
89 \( 1 + (-0.403 - 0.915i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−18.90172784462705370750772837108, −17.85356552483857416645857895370, −17.760507034555273539116644388094, −16.78268579838485328474195628882, −16.28620757875685303275891019461, −15.33986784966533473218504835345, −14.91905447309199250619159747557, −13.87654986659179743118620242864, −13.74232470291965283493067593504, −12.3637798849994498223326629142, −11.85659629471995772561285318211, −10.663408487316441684904707670487, −10.291191831032883011647480800598, −9.80596779217839169911304278420, −8.98461068244330195876526114219, −8.15139244363061141937052182791, −7.39238938955892433416934530261, −6.91258011588539735145305728501, −5.88664625829410135555209594271, −5.45191668382532442508371369481, −4.64393922490929228237155606636, −3.06431518816139615522308502583, −2.51849786738273271940580484389, −1.743906289545759204210126619526, −0.57605730860707764123491750413, 0.76093471201097308039754617, 1.65648457133981915412749274033, 2.53636877850686836686238772394, 2.973990849441699183861403829016, 4.36388262142673168666203504204, 4.99934120764633540797102356123, 6.01502297877121246071388956872, 6.717460636515426560691915805215, 7.65419394905747601211988187600, 8.23381513917408688546524393089, 9.08041392248913307360006744769, 9.58672907337440669791966642394, 10.484339997810892331700936449587, 10.64949085714553747972828882053, 11.927383873174079388873097703502, 12.40640599973011966353780668561, 13.096805350921858007532930853987, 13.7404242089770848885642959658, 14.71823579531539266377575235911, 15.60224736355701431315187264987, 16.18263527085419910446815026083, 17.03800063115061339242343908992, 17.45605958297648780333028449141, 17.981472705937411368486313104854, 18.79681590240172075080583786774

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