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Properties

Label 2-1815-165.8-c0-0-1
Degree $2$
Conductor $1815$
Sign $0.357 + 0.934i$
Analytic cond. $0.905802$
Root an. cond. $0.951736$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)3-s + (0.951 + 0.309i)4-s + (0.587 − 0.809i)5-s + (−0.809 + 0.587i)9-s − 0.999i·12-s + (−0.951 − 0.309i)15-s + (0.809 + 0.587i)16-s + (0.809 − 0.587i)20-s + (1 − i)23-s + (−0.309 − 0.951i)25-s + (0.809 + 0.587i)27-s + (−0.951 + 0.309i)36-s + (−0.642 − 1.26i)37-s + 0.999i·45-s + (−0.642 + 1.26i)47-s + (0.309 − 0.951i)48-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (0.951 + 0.309i)4-s + (0.587 − 0.809i)5-s + (−0.809 + 0.587i)9-s − 0.999i·12-s + (−0.951 − 0.309i)15-s + (0.809 + 0.587i)16-s + (0.809 − 0.587i)20-s + (1 − i)23-s + (−0.309 − 0.951i)25-s + (0.809 + 0.587i)27-s + (−0.951 + 0.309i)36-s + (−0.642 − 1.26i)37-s + 0.999i·45-s + (−0.642 + 1.26i)47-s + (0.309 − 0.951i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.357 + 0.934i$
Analytic conductor: \(0.905802\)
Root analytic conductor: \(0.951736\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :0),\ 0.357 + 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.396987491\)
\(L(\frac12)\) \(\approx\) \(1.396987491\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T^{2} \)
7 \( 1 + (0.587 - 0.809i)T^{2} \)
13 \( 1 + (-0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.642 + 1.26i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.221 + 1.39i)T + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.221 + 1.39i)T + (-0.951 - 0.309i)T^{2} \)
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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

−9.074793597049934375040214031773, −8.447416344513158361893446360926, −7.62515579130597417535550396115, −6.91266181205265003372544774462, −6.16894203880368351562429050819, −5.52224433438385018620035919780, −4.51611013859654070424605023613, −3.02442584421822000052884409048, −2.15268742741287376907190276604, −1.19104690188971072105960699319, 1.68943230532897609950812356435, 2.92698021271863933315648514283, 3.49390423163023580696949414755, 4.93943955916410051066993064203, 5.58496584353588588083068503658, 6.42807196787612654125694016257, 6.95605103481385696719953525949, 7.979401988566203713077987116387, 9.122363961944008694102826274451, 9.804017009294555108689083828482

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