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Properties

Label 2-465-31.5-c1-0-9
Degree $2$
Conductor $465$
Sign $0.893 + 0.449i$
Analytic cond. $3.71304$
Root an. cond. $1.92692$
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  − 2.09·2-s + (−0.5 + 0.866i)3-s + 2.37·4-s + (0.5 + 0.866i)5-s + (1.04 − 1.81i)6-s + (1.26 − 2.19i)7-s − 0.783·8-s + (−0.499 − 0.866i)9-s + (−1.04 − 1.81i)10-s + (1.41 + 2.45i)11-s + (−1.18 + 2.05i)12-s + (−2.49 − 4.32i)13-s + (−2.64 + 4.58i)14-s − 0.999·15-s − 3.11·16-s + (1.38 − 2.39i)17-s + ⋯
L(s)  = 1  − 1.47·2-s + (−0.288 + 0.499i)3-s + 1.18·4-s + (0.223 + 0.387i)5-s + (0.426 − 0.739i)6-s + (0.478 − 0.829i)7-s − 0.277·8-s + (−0.166 − 0.288i)9-s + (−0.330 − 0.572i)10-s + (0.427 + 0.740i)11-s + (−0.342 + 0.593i)12-s + (−0.692 − 1.19i)13-s + (−0.707 + 1.22i)14-s − 0.258·15-s − 0.777·16-s + (0.335 − 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.893 + 0.449i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (346, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.893 + 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.596968 - 0.141579i\)
\(L(\frac12)\) \(\approx\) \(0.596968 - 0.141579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-5.55 - 0.359i)T \)
good2 \( 1 + 2.09T + 2T^{2} \)
7 \( 1 + (-1.26 + 2.19i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.41 - 2.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.49 + 4.32i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.38 + 2.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.97 + 3.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.01T + 23T^{2} \)
29 \( 1 - 6.01T + 29T^{2} \)
37 \( 1 + (0.148 - 0.257i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.67 + 2.90i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.18 + 10.7i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.96T + 47T^{2} \)
53 \( 1 + (4.43 + 7.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.425 + 0.736i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 7.98T + 61T^{2} \)
67 \( 1 + (-3.84 - 6.65i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.59 - 2.76i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.74 - 9.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.67 + 9.83i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.66 + 2.88i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + 1.62T + 97T^{2} \)
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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

−10.41765408953554329683710938254, −10.18887642935191106780682185450, −9.438643243180996010661844169993, −8.283238223719568024142881455588, −7.48066093145257451913519091194, −6.77237101522581701580821087552, −5.29064633836834885225595908124, −4.16778976334694489639118153759, −2.46324444897665742995637949737, −0.74237106418213391596659400928, 1.26092323797266449380930507419, 2.30371618322738778345567315945, 4.41908284286668204349808312532, 5.81085837262342647013824706328, 6.60353392725328539669575698922, 7.925273909421087027509942900817, 8.313760152751996111691313027683, 9.290465749436606977479724237781, 9.949004067821733461649939921953, 11.02073485869335810547696438080

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