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Properties

Label 2-605-55.49-c1-0-8
Degree $2$
Conductor $605$
Sign $-0.857 - 0.515i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.763 − 0.248i)2-s + (−1.03 + 1.42i)3-s + (−1.09 + 0.796i)4-s + (1.42 + 1.71i)5-s + (−0.436 + 1.34i)6-s + (0.348 + 0.479i)7-s + (−1.58 + 2.17i)8-s + (−0.0309 − 0.0953i)9-s + (1.51 + 0.958i)10-s − 2.38i·12-s + (1.70 − 0.554i)13-s + (0.384 + 0.279i)14-s + (−3.92 + 0.256i)15-s + (0.169 − 0.522i)16-s + (−6.73 − 2.18i)17-s + (−0.0472 − 0.0650i)18-s + ⋯
L(s)  = 1  + (0.539 − 0.175i)2-s + (−0.597 + 0.822i)3-s + (−0.548 + 0.398i)4-s + (0.639 + 0.768i)5-s + (−0.178 + 0.548i)6-s + (0.131 + 0.181i)7-s + (−0.559 + 0.770i)8-s + (−0.0103 − 0.0317i)9-s + (0.479 + 0.302i)10-s − 0.689i·12-s + (0.473 − 0.153i)13-s + (0.102 + 0.0746i)14-s + (−1.01 + 0.0663i)15-s + (0.0424 − 0.130i)16-s + (−1.63 − 0.530i)17-s + (−0.0111 − 0.0153i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.857 - 0.515i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.857 - 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.321477 + 1.15920i\)
\(L(\frac12)\) \(\approx\) \(0.321477 + 1.15920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.42 - 1.71i)T \)
11 \( 1 \)
good2 \( 1 + (-0.763 + 0.248i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.03 - 1.42i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-0.348 - 0.479i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.70 + 0.554i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (6.73 + 2.18i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.85 - 1.34i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.49iT - 23T^{2} \)
29 \( 1 + (2.89 - 2.10i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.90 - 5.86i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.31 + 5.93i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.80 - 4.94i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.51iT - 43T^{2} \)
47 \( 1 + (-1.13 + 1.56i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.26 + 0.736i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.0309 - 0.0224i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.06 + 3.27i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 6.79iT - 67T^{2} \)
71 \( 1 + (3.64 - 11.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.01 + 5.51i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.39 - 4.30i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.65 - 1.83i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.21T + 89T^{2} \)
97 \( 1 + (-5.11 + 1.66i)T + (78.4 - 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.09752903297054908420573071393, −10.29193046939485004436407730152, −9.379109848180875672543480094375, −8.628193917642170425781413159721, −7.35612344699836628327514489050, −6.19746214740408053048826759464, −5.32317360151892604078933745100, −4.56380778743279501689006148071, −3.52755494078058099865472495110, −2.35059051770324192165269264817, 0.61298421803716422509177679568, 1.88013223124651688201112591216, 3.93995119744427791246005696520, 4.81795400806471345984086550901, 5.88371774263762681191201791941, 6.29076078111091218315658732961, 7.37676333172503903188770356858, 8.751647209003930977623112881525, 9.254700473116298114402714779622, 10.30134381261637352435229321503

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