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Properties

Label 2-325-65.49-c1-0-2
Degree $2$
Conductor $325$
Sign $-0.999 + 0.0416i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.999 + 0.0416i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.999 + 0.0416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0151936 - 0.729652i\)
\(L(\frac12)\) \(\approx\) \(0.0151936 - 0.729652i\)
\(L(\frac{3}{2})\) 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

−12.16171854407829582826553749524, −11.09201319044897608168643250886, −10.14061882544083246158458207805, −9.721115938808025298061140707803, −8.037988874568154826249967810226, −6.98240213983628104531551913791, −5.83661305679062093992434178585, −5.38340220083211607517104601049, −4.60247366845034528954822292871, −2.56332928278159905080505475020, 0.48926694816021345980827405187, 2.44067947875183660281909108774, 3.83039884949639367484794306516, 5.03503245878402742915104211276, 6.23682691565098616150058922140, 7.23765903461412041264627112561, 7.86541154967329565354770559507, 9.728884668497802727904321264429, 10.72574861650940805595798319629, 11.13230680675741548372673053557

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