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Properties

Label 2-325-65.7-c1-0-9
Degree $2$
Conductor $325$
Sign $-0.994 + 0.0999i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
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  + (−2.20 − 1.27i)2-s + (−0.473 − 1.76i)3-s + (2.24 + 3.88i)4-s + (−1.20 + 4.49i)6-s + (1.77 + 3.08i)7-s − 6.31i·8-s + (−0.294 + 0.170i)9-s + (−1.44 − 5.39i)11-s + (5.79 − 5.79i)12-s + (−2.23 − 2.82i)13-s − 9.05i·14-s + (−3.56 + 6.17i)16-s + (−0.522 − 0.140i)17-s + 0.865·18-s + (−3.16 − 0.848i)19-s + ⋯
L(s)  = 1  + (−1.55 − 0.900i)2-s + (−0.273 − 1.01i)3-s + (1.12 + 1.94i)4-s + (−0.491 + 1.83i)6-s + (0.672 + 1.16i)7-s − 2.23i·8-s + (−0.0981 + 0.0566i)9-s + (−0.435 − 1.62i)11-s + (1.67 − 1.67i)12-s + (−0.619 − 0.784i)13-s − 2.42i·14-s + (−0.890 + 1.54i)16-s + (−0.126 − 0.0339i)17-s + 0.204·18-s + (−0.726 − 0.194i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.0999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.0999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.994 + 0.0999i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.994 + 0.0999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0235552 - 0.469924i\)
\(L(\frac12)\) \(\approx\) \(0.0235552 - 0.469924i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (2.23 + 2.82i)T \)
good2 \( 1 + (2.20 + 1.27i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.473 + 1.76i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.77 - 3.08i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.44 + 5.39i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.522 + 0.140i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.16 + 0.848i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.62 + 1.50i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.795 - 0.459i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.614 + 0.614i)T + 31iT^{2} \)
37 \( 1 + (-2.52 + 4.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.34 - 1.69i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.46 + 5.48i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + (-1.91 + 1.91i)T - 53iT^{2} \)
59 \( 1 + (0.0766 - 0.286i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.24 - 2.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.17 - 3.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.36 + 12.5i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 3.87iT - 73T^{2} \)
79 \( 1 + 8.77iT - 79T^{2} \)
83 \( 1 + 5.05T + 83T^{2} \)
89 \( 1 + (-0.609 + 0.163i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.55 - 1.47i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.19514972512364371131329212320, −10.37235556142087184367105134180, −9.102504232227984298394672596031, −8.410608548226452068456235416768, −7.79483949925727006878334664690, −6.63396073794739445589359092939, −5.39646338737842960166760407378, −3.04436807842147361406274808027, −2.04403379183405111757378347103, −0.57811735555598214003565841136, 1.71419582886470126005539073396, 4.39224928520321895173725512204, 5.05775365776969720500714233476, 6.79027116771597807169979387902, 7.31502182205135215249886732402, 8.260673957270173246127477108355, 9.523188846002173135968158022022, 9.942392039654005812528807876206, 10.66935894523908949220734672637, 11.39469672865141581465891262910

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