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Properties

Label 2-605-55.17-c1-0-28
Degree $2$
Conductor $605$
Sign $0.281 - 0.959i$
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  + (1.99 + 1.01i)2-s + (1.39 − 0.221i)3-s + (1.76 + 2.42i)4-s + (0.333 + 2.21i)5-s + (3.00 + 0.977i)6-s + (0.349 + 2.20i)8-s + (−0.951 + 0.309i)9-s + (−1.58 + 4.74i)10-s + (3.00 + 3i)12-s + (2.03 − 3.98i)13-s + (0.954 + 3.01i)15-s + (0.309 − 0.951i)16-s + (2.03 + 3.98i)17-s + (−2.20 − 0.349i)18-s + (−5.11 − 3.71i)19-s + (−4.77 + 4.70i)20-s + ⋯
L(s)  = 1  + (1.40 + 0.717i)2-s + (0.806 − 0.127i)3-s + (0.881 + 1.21i)4-s + (0.148 + 0.988i)5-s + (1.22 + 0.398i)6-s + (0.123 + 0.780i)8-s + (−0.317 + 0.103i)9-s + (−0.499 + 1.50i)10-s + (0.866 + 0.866i)12-s + (0.563 − 1.10i)13-s + (0.246 + 0.778i)15-s + (0.0772 − 0.237i)16-s + (0.492 + 0.966i)17-s + (−0.520 − 0.0824i)18-s + (−1.17 − 0.852i)19-s + (−1.06 + 1.05i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.281 - 0.959i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.281 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.10016 + 2.32178i\)
\(L(\frac12)\) \(\approx\) \(3.10016 + 2.32178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.333 - 2.21i)T \)
11 \( 1 \)
good2 \( 1 + (-1.99 - 1.01i)T + (1.17 + 1.61i)T^{2} \)
3 \( 1 + (-1.39 + 0.221i)T + (2.85 - 0.927i)T^{2} \)
7 \( 1 + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-2.03 + 3.98i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-2.03 - 3.98i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (5.11 + 3.71i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 + (-5.11 + 3.71i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.618 - 1.90i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.19 + 0.663i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (-3.71 + 5.11i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (0.663 + 4.19i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (1.26 + 0.642i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-3.52 - 4.85i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (6.01 + 1.95i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 + (2.47 - 7.60i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.41 + 0.699i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (1.95 + 6.01i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (7.96 - 4.06i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + (-4.49 + 8.82i)T + (-57.0 - 78.4i)T^{2} \)
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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

−10.86132297266248728450529756423, −10.15866781089100659460250055314, −8.723391895354195522003659177845, −7.953389938389426342116958452407, −7.09311376405669487033257719124, −6.15331873781882931298006764541, −5.53925136182474008384657769337, −4.11311881706733152144585870231, −3.26590900260675372155979768762, −2.44923122500692571851354851557, 1.64962664635718905252643379638, 2.76373053940226723946194492509, 3.88034057115411988173869931542, 4.55471519561285252350547597046, 5.60254978027743720515878020104, 6.48061531355460756582574816427, 8.084338293895072288159642167218, 8.781653908467507800720482646493, 9.606388374678806100915795543543, 10.69397785646578118998158910509

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