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Properties

Label 2-1575-105.83-c0-0-6
Degree $2$
Conductor $1575$
Sign $-0.749 + 0.662i$
Analytic cond. $0.786027$
Root an. cond. $0.886581$
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  + (−1 − i)2-s + i·4-s + (−0.707 + 0.707i)7-s − 1.41i·11-s + 1.41·14-s + 16-s + (−1.41 + 1.41i)22-s + (1 − i)23-s + (−0.707 − 0.707i)28-s − 1.41·29-s + (−1 − i)32-s + (1.41 − 1.41i)37-s + (−1.41 − 1.41i)43-s + 1.41·44-s − 2·46-s + ⋯
L(s)  = 1  + (−1 − i)2-s + i·4-s + (−0.707 + 0.707i)7-s − 1.41i·11-s + 1.41·14-s + 16-s + (−1.41 + 1.41i)22-s + (1 − i)23-s + (−0.707 − 0.707i)28-s − 1.41·29-s + (−1 − i)32-s + (1.41 − 1.41i)37-s + (−1.41 − 1.41i)43-s + 1.41·44-s − 2·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.749 + 0.662i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (818, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.749 + 0.662i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4799929473\)
\(L(\frac12)\) \(\approx\) \(0.4799929473\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1 + i)T + iT^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.242454152782392599432558068946, −8.809406131187633599072289804249, −8.174510003257792654563353136146, −7.03389790411142241419526366732, −5.98411915569158143543096113180, −5.38058519732826601623193750417, −3.75531217525883219134556207398, −2.98147865358503158802669113073, −2.11365510750594449218689920851, −0.56453713107814773700230536141, 1.35851116374147052679949178726, 3.03114450872281549054864803105, 4.12497499643371629024693422102, 5.19765565831339564062725644564, 6.26832763004306172893859986984, 6.94340333399094639314802054835, 7.48490087090307956312785115857, 8.134487668699395820375558157312, 9.382041697360766707062629613688, 9.527287496820352143026150131037

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