Properties

Label 2-325-65.49-c1-0-12
Degree $2$
Conductor $325$
Sign $0.743 - 0.668i$
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  + (0.866 + 1.5i)2-s + (1.73 − i)3-s + (−0.5 + 0.866i)4-s + (3 + 1.73i)6-s + 1.73·8-s + (0.499 − 0.866i)9-s + 2i·12-s + (−2.59 − 2.5i)13-s + (2.49 + 4.33i)16-s + (−2.59 − 1.5i)17-s + 1.73·18-s + (3 + 1.73i)19-s + (−5.19 + 3i)23-s + (2.99 − 1.73i)24-s + (1.5 − 6.06i)26-s + 4.00i·27-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + (0.999 − 0.577i)3-s + (−0.250 + 0.433i)4-s + (1.22 + 0.707i)6-s + 0.612·8-s + (0.166 − 0.288i)9-s + 0.577i·12-s + (−0.720 − 0.693i)13-s + (0.624 + 1.08i)16-s + (−0.630 − 0.363i)17-s + 0.408·18-s + (0.688 + 0.397i)19-s + (−1.08 + 0.625i)23-s + (0.612 − 0.353i)24-s + (0.294 − 1.18i)26-s + 0.769i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.743 - 0.668i$
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: \((2,\ 325,\ (\ :1/2),\ 0.743 - 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26819 + 0.869143i\)
\(L(\frac12)\) \(\approx\) \(2.26819 + 0.869143i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 + 2.5i)T \)
good2 \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.73 + i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.59 + 1.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (4.33 + 7.5i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.92 + 4i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + (6 + 3.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.73 + 3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.73T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.46 + 6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.06320582140642194343127519959, −10.72994803323307970102787496361, −9.657761828461666111533494596704, −8.478441774728160374487031605972, −7.65626478395584526847856818861, −7.11547378560333799605695580596, −5.87045527208178018879509423760, −4.91286322915608057445114060851, −3.47754814247966256008671545469, −2.01191929565435238423376921217, 2.05680907530095037741127653522, 3.07299183284244559893352566546, 4.06613634027397434960083526063, 4.91454422028328880442707367052, 6.66527761467482812661424846630, 7.929283025206573495705305189137, 8.896008532303529475038143230164, 9.867664529973238749599914886616, 10.50331212733602565269192500796, 11.75427174722118970411160632633

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