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Properties

Label 2-405-135.122-c1-0-6
Degree $2$
Conductor $405$
Sign $-0.331 - 0.943i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
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  + (2.18 + 1.52i)2-s + (1.73 + 4.77i)4-s + (−1.64 − 1.51i)5-s + (1.30 + 2.78i)7-s + (−2.12 + 7.92i)8-s + (−1.28 − 5.81i)10-s + (0.426 + 0.508i)11-s + (−0.269 − 0.384i)13-s + (−1.42 + 8.06i)14-s + (−8.94 + 7.50i)16-s + (−1.20 − 4.50i)17-s + (4.80 − 2.77i)19-s + (4.34 − 10.5i)20-s + (0.154 + 1.76i)22-s + (0.0602 + 0.0280i)23-s + ⋯
L(s)  = 1  + (1.54 + 1.07i)2-s + (0.869 + 2.38i)4-s + (−0.737 − 0.675i)5-s + (0.491 + 1.05i)7-s + (−0.750 + 2.80i)8-s + (−0.407 − 1.83i)10-s + (0.128 + 0.153i)11-s + (−0.0746 − 0.106i)13-s + (−0.380 + 2.15i)14-s + (−2.23 + 1.87i)16-s + (−0.292 − 1.09i)17-s + (1.10 − 0.637i)19-s + (0.972 − 2.34i)20-s + (0.0328 + 0.375i)22-s + (0.0125 + 0.00585i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.331 - 0.943i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66186 + 2.34605i\)
\(L(\frac12)\) \(\approx\) \(1.66186 + 2.34605i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.64 + 1.51i)T \)
good2 \( 1 + (-2.18 - 1.52i)T + (0.684 + 1.87i)T^{2} \)
7 \( 1 + (-1.30 - 2.78i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (-0.426 - 0.508i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.269 + 0.384i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (1.20 + 4.50i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-4.80 + 2.77i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0602 - 0.0280i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.434 + 2.46i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.76 + 0.642i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (2.44 - 0.656i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.62 + 0.286i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.732 + 8.36i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-3.71 + 1.73i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (7.52 + 7.52i)T + 53iT^{2} \)
59 \( 1 + (-3.49 - 2.93i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (5.84 + 2.12i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (4.48 - 3.13i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-5.33 - 3.07i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-13.6 - 3.64i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.34 - 0.941i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (3.25 - 4.64i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (-3.08 - 5.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.5 + 1.18i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.86996905399605425964482214695, −11.34548548163029308243137493582, −9.295116873170174149617365767046, −8.417627947138894565559342500776, −7.59332599156702575667137336630, −6.74414603358114757987878833911, −5.34791415791044157847094926186, −5.05767292715144592318576284610, −3.89789805854824542631945508793, −2.64690700998015598573477744532, 1.43163658792941009953552947446, 3.06092208227692981404136201494, 3.90154662152656348097049963388, 4.65283950859573844231999708278, 5.94977980629238301174746861706, 6.94723858858363522877646588867, 7.979216235079794447420594496673, 9.713475800888984100365190774942, 10.74738226297253276426358648490, 10.94651734947672233096711297708

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