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Properties

Label 1-1412-1412.1351-r1-0-0
Degree $1$
Conductor $1412$
Sign $-0.968 - 0.248i$
Analytic cond. $151.740$
Root an. cond. $151.740$
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  + (−0.997 + 0.0713i)3-s + (−0.997 + 0.0713i)5-s + (−0.707 + 0.707i)7-s + (0.989 − 0.142i)9-s + (0.415 − 0.909i)11-s + (0.479 + 0.877i)13-s + (0.989 − 0.142i)15-s + (−0.841 + 0.540i)17-s + (−0.281 − 0.959i)19-s + (0.654 − 0.755i)21-s + (−0.909 + 0.415i)23-s + (0.989 − 0.142i)25-s + (−0.977 + 0.212i)27-s + (0.142 + 0.989i)29-s + (−0.936 + 0.349i)31-s + ⋯
L(s)  = 1  + (−0.997 + 0.0713i)3-s + (−0.997 + 0.0713i)5-s + (−0.707 + 0.707i)7-s + (0.989 − 0.142i)9-s + (0.415 − 0.909i)11-s + (0.479 + 0.877i)13-s + (0.989 − 0.142i)15-s + (−0.841 + 0.540i)17-s + (−0.281 − 0.959i)19-s + (0.654 − 0.755i)21-s + (−0.909 + 0.415i)23-s + (0.989 − 0.142i)25-s + (−0.977 + 0.212i)27-s + (0.142 + 0.989i)29-s + (−0.936 + 0.349i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $-0.968 - 0.248i$
Analytic conductor: \(151.740\)
Root analytic conductor: \(151.740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1412,\ (1:\ ),\ -0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03300342881 + 0.2618889895i\)
\(L(\frac12)\) \(\approx\) \(-0.03300342881 + 0.2618889895i\)
\(L(1)\) \(\approx\) \(0.5502371236 + 0.1097962608i\)
\(L(1)\) \(\approx\) \(0.5502371236 + 0.1097962608i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
353 \( 1 \)
good3 \( 1 + (-0.997 + 0.0713i)T \)
5 \( 1 + (-0.997 + 0.0713i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (0.415 - 0.909i)T \)
13 \( 1 + (0.479 + 0.877i)T \)
17 \( 1 + (-0.841 + 0.540i)T \)
19 \( 1 + (-0.281 - 0.959i)T \)
23 \( 1 + (-0.909 + 0.415i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (-0.936 + 0.349i)T \)
37 \( 1 + (0.212 - 0.977i)T \)
41 \( 1 + (-0.909 - 0.415i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (0.989 + 0.142i)T \)
53 \( 1 + (0.0713 + 0.997i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (0.599 + 0.800i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (0.997 + 0.0713i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (-0.212 - 0.977i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−20.356834461628093568974921490867, −19.41514013840475918568010256956, −18.60444754006914875957632145972, −17.91837138558188287197214988535, −17.02353068582726088321223687857, −16.49203599879511467847330331707, −15.64528845196039675755593359521, −15.24159420652481268419096057838, −13.99812266813076087960986618817, −12.99871878644958310328132794269, −12.485447470029376073817115494494, −11.751114474904045646267801479888, −10.9633801239688956115253279450, −10.23463330064090305363554658170, −9.55419479643750442711317247741, −8.23322624248257351798445303501, −7.50510810711275642167395472638, −6.73439325634354977730382476966, −6.08300185464042834288023798992, −4.91154686502777682641184716568, −4.1238078533007241174275556256, −3.5634834081266577813917896288, −2.04332655834111528433515695722, −0.74643242731461616653357529070, −0.101952067977827259544140897328, 0.932268468384258001641669518476, 2.244224187980133519706627778565, 3.59768356887097293678065915843, 4.079363865890022365757173369359, 5.16760641954739807026964895282, 6.12655069615583497792696175559, 6.64519271125696519390625354811, 7.49574419702597093028063640781, 8.885644239680652428050975878565, 9.034714618609232695253461888574, 10.50325828833810145962744538279, 11.11297389816787820728075186798, 11.70255146819383617233552951691, 12.42391488634779417562214104848, 13.12080216555483062977934303763, 14.16356684838618380750363538800, 15.25401828376512791609854846760, 15.90118238262761178256276832451, 16.25910214821023322113513528012, 17.10167855515326881559857806054, 18.10831384296155469757338815330, 18.70868171879954489891365294723, 19.426554173795284436500450435763, 19.97642861614049824924925427385, 21.33325953117454094596187448527

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