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Properties

Label 2-605-55.14-c1-0-11
Degree $2$
Conductor $605$
Sign $-0.858 + 0.513i$
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  + (1.48 + 2.04i)2-s + (−0.753 + 0.244i)3-s + (−1.35 + 4.15i)4-s + (−1.80 + 1.31i)5-s + (−1.61 − 1.17i)6-s + (3.29 + 1.07i)7-s + (−5.69 + 1.85i)8-s + (−1.91 + 1.39i)9-s + (−5.37 − 1.73i)10-s − 3.46i·12-s + (2.70 + 8.31i)14-s + (1.03 − 1.43i)15-s + (−5.15 − 3.74i)16-s + (0.931 − 1.28i)17-s + (−5.69 − 1.85i)18-s + (−1.23 − 3.80i)19-s + ⋯
L(s)  = 1  + (1.04 + 1.44i)2-s + (−0.435 + 0.141i)3-s + (−0.675 + 2.07i)4-s + (−0.807 + 0.589i)5-s + (−0.660 − 0.479i)6-s + (1.24 + 0.404i)7-s + (−2.01 + 0.654i)8-s + (−0.639 + 0.464i)9-s + (−1.69 − 0.547i)10-s − 0.999i·12-s + (0.722 + 2.22i)14-s + (0.268 − 0.370i)15-s + (−1.28 − 0.936i)16-s + (0.225 − 0.310i)17-s + (−1.34 − 0.436i)18-s + (−0.283 − 0.872i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.858 + 0.513i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.858 + 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461369 - 1.67005i\)
\(L(\frac12)\) \(\approx\) \(0.461369 - 1.67005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.80 - 1.31i)T \)
11 \( 1 \)
good2 \( 1 + (-1.48 - 2.04i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.753 - 0.244i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-3.29 - 1.07i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.931 + 1.28i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.23 + 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.792iT - 23T^{2} \)
29 \( 1 + (2.70 - 8.31i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.72 - 1.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.03 + 0.335i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.70 - 8.31i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (-6.30 + 2.04i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.93 - 8.16i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.27 + 7.01i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.602 - 0.437i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 9.30iT - 67T^{2} \)
71 \( 1 + (-8.18 - 5.94i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.58 - 2.14i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.01 + 0.737i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.89 - 5.36i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 + (3.43 + 4.72i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.32837387427112147371624596665, −10.70101543613684959554167855014, −8.838905883823067803856793889060, −8.210524776197548948169320283643, −7.41766143152855810292328342534, −6.68082458140520145197528176649, −5.49298430190180678097670844447, −4.99917258998100312043390401042, −4.05480792705785020417001558250, −2.76413225957731913798558968228, 0.74939866285875924970492659256, 1.99940728107360899920970340015, 3.60937310557964884744266094379, 4.24484374470283367171421623220, 5.21305984850961839076617805261, 5.94428991585294720242660165249, 7.57830855722491542602952338131, 8.478169461053225166123907556615, 9.573891805690852359643949011231, 10.73910767057184856477538081207

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