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Properties

Label 2-325-65.4-c1-0-7
Degree $2$
Conductor $325$
Sign $-0.566 - 0.824i$
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  + (−0.609 + 1.05i)2-s + (2.01 + 1.16i)3-s + (0.256 + 0.443i)4-s + (−2.46 + 1.42i)6-s + (1.80 + 3.11i)7-s − 3.06·8-s + (1.21 + 2.11i)9-s + (−4.65 − 2.68i)11-s + 1.19i·12-s + (3.11 − 1.81i)13-s − 4.39·14-s + (1.35 − 2.34i)16-s + (0.980 − 0.565i)17-s − 2.97·18-s + (1.96 − 1.13i)19-s + ⋯
L(s)  = 1  + (−0.431 + 0.746i)2-s + (1.16 + 0.673i)3-s + (0.128 + 0.221i)4-s + (−1.00 + 0.580i)6-s + (0.680 + 1.17i)7-s − 1.08·8-s + (0.406 + 0.704i)9-s + (−1.40 − 0.809i)11-s + 0.344i·12-s + (0.863 − 0.504i)13-s − 1.17·14-s + (0.339 − 0.587i)16-s + (0.237 − 0.137i)17-s − 0.701·18-s + (0.450 − 0.260i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.566 - 0.824i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.566 - 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738067 + 1.40236i\)
\(L(\frac12)\) \(\approx\) \(0.738067 + 1.40236i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.11 + 1.81i)T \)
good2 \( 1 + (0.609 - 1.05i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-2.01 - 1.16i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.80 - 3.11i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.65 + 2.68i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.980 + 0.565i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.96 + 1.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.37 - 1.94i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0123 - 0.0214i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.46iT - 31T^{2} \)
37 \( 1 + (4.35 - 7.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.23 - 1.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.980 - 0.565i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.58T + 47T^{2} \)
53 \( 1 - 4.43iT - 53T^{2} \)
59 \( 1 + (-0.148 + 0.0857i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.68 + 2.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.19 - 5.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.35 + 5.39i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.70T + 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + (13.9 + 8.07i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.08 + 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.80741504058654031193252214628, −10.92808715361215287701781289666, −9.656697205518000761915378124216, −8.768826361983389311379825680614, −8.277803357838135152084780694654, −7.66851513951232085931498653370, −6.03834459674527141825080692774, −5.14468485952559807497901146419, −3.32840302596671025132987186616, −2.64600453726787908718063424784, 1.31675644986021244508792754497, 2.35492901186046415991860322158, 3.60016905542748270487346796398, 5.16947345632594058252000484142, 6.83762160779332785955821167541, 7.64807129440098936787424896371, 8.468378730741990719009748056148, 9.449056881657504218059249304830, 10.55794773753927168680921474682, 10.91315095210119564030602202791

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