Properties

Label 2-605-11.9-c1-0-33
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: $\chi_{605} (251, \cdot )$
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

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

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