↑

Properties

Label 1-373-373.68-r0-0-0
Degree $1$
Conductor $373$
Sign $0.305 + 0.952i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.557 + 0.830i)2-s + (0.780 − 0.625i)3-s + (−0.378 − 0.925i)4-s + (0.585 + 0.810i)5-s + (0.0843 + 0.996i)6-s + (0.688 + 0.724i)7-s + (0.979 + 0.201i)8-s + (0.217 − 0.975i)9-s + (−0.999 + 0.0337i)10-s + (−0.315 + 0.948i)11-s + (−0.874 − 0.485i)12-s + (0.979 − 0.201i)13-s + (−0.985 + 0.168i)14-s + (0.963 + 0.266i)15-s + (−0.713 + 0.701i)16-s + (−0.612 + 0.790i)17-s + ⋯
L(s)  = 1  + (−0.557 + 0.830i)2-s + (0.780 − 0.625i)3-s + (−0.378 − 0.925i)4-s + (0.585 + 0.810i)5-s + (0.0843 + 0.996i)6-s + (0.688 + 0.724i)7-s + (0.979 + 0.201i)8-s + (0.217 − 0.975i)9-s + (−0.999 + 0.0337i)10-s + (−0.315 + 0.948i)11-s + (−0.874 − 0.485i)12-s + (0.979 − 0.201i)13-s + (−0.985 + 0.168i)14-s + (0.963 + 0.266i)15-s + (−0.713 + 0.701i)16-s + (−0.612 + 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.305 + 0.952i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.305 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.174713814 + 0.8567806933i\)
\(L(\frac12)\) \(\approx\) \(1.174713814 + 0.8567806933i\)
\(L(1)\) \(\approx\) \(1.088882870 + 0.4520223639i\)
\(L(1)\) \(\approx\) \(1.088882870 + 0.4520223639i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.557 + 0.830i)T \)
3 \( 1 + (0.780 - 0.625i)T \)
5 \( 1 + (0.585 + 0.810i)T \)
7 \( 1 + (0.688 + 0.724i)T \)
11 \( 1 + (-0.315 + 0.948i)T \)
13 \( 1 + (0.979 - 0.201i)T \)
17 \( 1 + (-0.612 + 0.790i)T \)
19 \( 1 + (-0.250 + 0.968i)T \)
23 \( 1 + (0.151 - 0.988i)T \)
29 \( 1 + (-0.184 - 0.982i)T \)
31 \( 1 + (-0.874 + 0.485i)T \)
37 \( 1 + (-0.801 + 0.598i)T \)
41 \( 1 + (0.820 - 0.571i)T \)
43 \( 1 + (-0.378 - 0.925i)T \)
47 \( 1 + (0.638 - 0.769i)T \)
53 \( 1 + (0.217 + 0.975i)T \)
59 \( 1 + (0.943 + 0.331i)T \)
61 \( 1 + (0.780 - 0.625i)T \)
67 \( 1 + (-0.994 + 0.101i)T \)
71 \( 1 + (0.990 - 0.134i)T \)
73 \( 1 + (0.736 - 0.676i)T \)
79 \( 1 + (-0.999 + 0.0337i)T \)
83 \( 1 + (0.217 + 0.975i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.612 + 0.790i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.58418045604371090509847646017, −23.67943856043756357240817227927, −22.22272652685275092309974899685, −21.311399586721321877410341589186, −20.96568924639482452189120459996, −20.149872081289280685964977593657, −19.50230284944073054174606696965, −18.280402336980649816597454841386, −17.49921897278424402195365250762, −16.43517515470616693163153766066, −15.92909022666293501281050151170, −14.27590276312439687482138991834, −13.46696972909932375441479225722, −13.08448993711431066330252137066, −11.26059112674369530232302124776, −10.93124238380243260123890469373, −9.70227816721671766230606114901, −8.91589452609567824849947759013, −8.36370520750927462387169558298, −7.25353256316827338708430089154, −5.30894480634028656780219701385, −4.37618572619535565420077196583, −3.41611300933825735261946517884, −2.17852565287886416966448737451, −1.073459511686345290219813405482, 1.66544273700150985128492556176, 2.2841849936242766498068888240, 3.9915655955623976531140602401, 5.56027233474855282463048201414, 6.38819899171119106570783982955, 7.27746774957315116519880134641, 8.274876813151181529009002469313, 8.89776013497585534978944327753, 10.054934936079564370786370958554, 10.883166942510785681567178529630, 12.3976837978717553136951436723, 13.45085281173114942661043159401, 14.31055436217176539788891213817, 15.026326804011466014334193732767, 15.523377974659992659592683398781, 17.148747386318413031185054240339, 17.9483057630537422323316131773, 18.465386681579665624106598479956, 19.07237680980058787191289257800, 20.350682626206292487617449966932, 21.09837288948600424871312457661, 22.41250907237973830118735607732, 23.2846478542913543425028482919, 24.17873818185362549961459732335, 25.10715572020569460143409336591

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