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Properties

Label 2-605-11.5-c1-0-13
Degree $2$
Conductor $605$
Sign $-0.998 + 0.0475i$
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  + (0.746 + 2.29i)2-s + (2.28 + 1.66i)3-s + (−3.09 + 2.25i)4-s + (−0.309 + 0.951i)5-s + (−2.11 + 6.49i)6-s + (1.61 − 1.17i)7-s + (−3.57 − 2.59i)8-s + (1.54 + 4.75i)9-s − 2.41·10-s − 10.8·12-s + (−0.362 − 1.11i)13-s + (3.90 + 2.83i)14-s + (−2.28 + 1.66i)15-s + (0.927 − 2.85i)16-s + (2.11 − 6.49i)17-s + (−9.76 + 7.09i)18-s + ⋯
L(s)  = 1  + (0.527 + 1.62i)2-s + (1.32 + 0.959i)3-s + (−1.54 + 1.12i)4-s + (−0.138 + 0.425i)5-s + (−0.861 + 2.65i)6-s + (0.611 − 0.444i)7-s + (−1.26 − 0.917i)8-s + (0.515 + 1.58i)9-s − 0.763·10-s − 3.12·12-s + (−0.100 − 0.309i)13-s + (1.04 + 0.758i)14-s + (−0.590 + 0.429i)15-s + (0.231 − 0.713i)16-s + (0.511 − 1.57i)17-s + (−2.30 + 1.67i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.998 + 0.0475i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.998 + 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0663317 - 2.78957i\)
\(L(\frac12)\) \(\approx\) \(0.0663317 - 2.78957i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.746 - 2.29i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-2.28 - 1.66i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-1.61 + 1.17i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.362 + 1.11i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.11 + 6.49i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + (-2.95 + 2.14i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.19 + 4.50i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.85 + 3.52i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (2.28 + 1.66i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.60 - 11.0i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.34 - 0.973i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.87 - 8.85i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 + (-3.49 + 10.7i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.947 + 0.688i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.23 - 3.80i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.85 - 5.70i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (-1.13 - 3.47i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.84669510101721524338160086111, −9.864049491387276529192277307693, −9.109213869149969928841890482189, −8.102788600110039043792197988945, −7.72037461902932737252790326374, −6.82951456435148881261071490044, −5.47692122419752596798616554586, −4.60405530341096941107294986585, −3.85388557171537452857559048606, −2.74125642281053757565617246712, 1.40182535515405329537560141060, 2.04917137861663015368768694932, 3.17181241437760277068652452318, 4.05838128042875065943048126888, 5.20951618095426927692875477852, 6.62308283868998504790329286726, 8.156897811836667373381487953260, 8.349397752894488673554007597368, 9.464910972175175096435970781492, 10.21018459055156808088866454424

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