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Properties

Label 2-325-65.49-c1-0-0
Degree $2$
Conductor $325$
Sign $-0.0192 - 0.999i$
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  + (−0.747 − 1.29i)2-s + (0.0820 − 0.0473i)3-s + (−0.118 + 0.204i)4-s + (−0.122 − 0.0708i)6-s + (−2.41 + 4.18i)7-s − 2.63·8-s + (−1.49 + 2.59i)9-s + (−0.926 + 0.534i)11-s + 0.0224i·12-s + (−3.59 + 0.331i)13-s + 7.21·14-s + (2.20 + 3.82i)16-s + (−3.08 − 1.77i)17-s + 4.47·18-s + (−4.96 − 2.86i)19-s + ⋯
L(s)  = 1  + (−0.528 − 0.915i)2-s + (0.0473 − 0.0273i)3-s + (−0.0591 + 0.102i)4-s + (−0.0501 − 0.0289i)6-s + (−0.912 + 1.57i)7-s − 0.932·8-s + (−0.498 + 0.863i)9-s + (−0.279 + 0.161i)11-s + 0.00647i·12-s + (−0.995 + 0.0918i)13-s + 1.92·14-s + (0.552 + 0.956i)16-s + (−0.747 − 0.431i)17-s + 1.05·18-s + (−1.13 − 0.657i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0192 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0192 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.0192 - 0.999i$
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: \((2,\ 325,\ (\ :1/2),\ -0.0192 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.201810 + 0.205733i\)
\(L(\frac12)\) \(\approx\) \(0.201810 + 0.205733i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 - 0.331i)T \)
good2 \( 1 + (0.747 + 1.29i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.0820 + 0.0473i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (2.41 - 4.18i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.926 - 0.534i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.08 + 1.77i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.96 + 2.86i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.13 + 3.54i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.736 - 1.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (-0.0126 - 0.0219i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.232 - 0.133i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.08 - 1.77i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.51T + 47T^{2} \)
53 \( 1 - 0.991iT - 53T^{2} \)
59 \( 1 + (-7.55 - 4.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.16 - 5.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.58 - 4.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.72 + 3.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + 8.78T + 79T^{2} \)
83 \( 1 + 0.725T + 83T^{2} \)
89 \( 1 + (11.6 - 6.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.71 - 2.97i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.72944597282087941952314025154, −10.89403421177617214149930913211, −10.05877106914081545200409336558, −8.960364255225740682964244216142, −8.732979981814660639389091693684, −7.00731649940359304144082884737, −5.92525454820273178811926992251, −4.88764732345470730006271471301, −2.64380311575067098626182506105, −2.48587165078301771644247764813, 0.21855279440746015151208014677, 3.03314746885629165461558401851, 4.13600969841340959498588193098, 5.87067518277733001575446211769, 6.78810825224150676491745629061, 7.34510551499835348352522743641, 8.449225466181464344322921661852, 9.383188278595456491773358124164, 10.20444174896713096554819968125, 11.23018334535469063919420956140

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