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Properties

Label 2-475-19.7-c1-0-12
Degree $2$
Conductor $475$
Sign $-0.981 - 0.189i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
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  + (1.08 + 1.87i)2-s + (1.47 + 2.55i)3-s + (−1.35 + 2.34i)4-s + (−3.20 + 5.54i)6-s − 0.591·7-s − 1.53·8-s + (−2.85 + 4.94i)9-s + 2.58·11-s − 7.99·12-s + (3.43 − 5.94i)13-s + (−0.641 − 1.11i)14-s + (1.03 + 1.79i)16-s + (−2.61 − 4.53i)17-s − 12.3·18-s + (−2.26 − 3.72i)19-s + ⋯
L(s)  = 1  + (0.767 + 1.32i)2-s + (0.852 + 1.47i)3-s + (−0.677 + 1.17i)4-s + (−1.30 + 2.26i)6-s − 0.223·7-s − 0.544·8-s + (−0.952 + 1.64i)9-s + 0.778·11-s − 2.30·12-s + (0.952 − 1.64i)13-s + (−0.171 − 0.297i)14-s + (0.259 + 0.449i)16-s + (−0.634 − 1.09i)17-s − 2.92·18-s + (−0.519 − 0.854i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.981 - 0.189i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.981 - 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256794 + 2.69146i\)
\(L(\frac12)\) \(\approx\) \(0.256794 + 2.69146i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (2.26 + 3.72i)T \)
good2 \( 1 + (-1.08 - 1.87i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.47 - 2.55i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 0.591T + 7T^{2} \)
11 \( 1 - 2.58T + 11T^{2} \)
13 \( 1 + (-3.43 + 5.94i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.61 + 4.53i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.45 + 2.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.52 - 6.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.81T + 31T^{2} \)
37 \( 1 - 4.82T + 37T^{2} \)
41 \( 1 + (3.11 + 5.39i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.18 - 3.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.27 + 2.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.79 - 8.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.46 - 2.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.16 - 2.01i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.15 + 3.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.74 - 11.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.21 + 7.29i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.93 + 5.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.02T + 83T^{2} \)
89 \( 1 + (1.85 - 3.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.26 - 2.18i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.09246963696288882341309426112, −10.53197525102655686383753198835, −9.237301086008468488991451618615, −8.781876801359803646136799270714, −7.79354566619000198715250856038, −6.73307789951198154064492312507, −5.59115672517596570753913257860, −4.79272293951310965207754900199, −3.87086011278252867237573244936, −3.01032154388792757433379296117, 1.55886128742194172565913182763, 2.01877661776619515883369938187, 3.51782505641970141088888726531, 4.13983077620309079161761631065, 6.05928327172089374529583630387, 6.74232659714767712862559013074, 7.939430841861755470862365578477, 8.897005524381919266349680503928, 9.670808968296713867517520471557, 11.14407586293573320507040789856

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