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Properties

Label 2-1260-105.2-c1-0-6
Degree $2$
Conductor $1260$
Sign $0.996 - 0.0892i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  + (−2.03 + 0.918i)5-s + (1.31 − 2.29i)7-s + (−5.56 + 3.21i)11-s + (1.53 − 1.53i)13-s + (3.67 + 0.985i)17-s + (−1.19 − 0.687i)19-s + (8.15 − 2.18i)23-s + (3.31 − 3.74i)25-s + 5.95·29-s + (4.97 + 8.61i)31-s + (−0.566 + 5.88i)35-s + (−2.39 + 0.642i)37-s − 1.18i·41-s + (3.28 − 3.28i)43-s + (−0.655 − 2.44i)47-s + ⋯
L(s)  = 1  + (−0.911 + 0.410i)5-s + (0.495 − 0.868i)7-s + (−1.67 + 0.967i)11-s + (0.425 − 0.425i)13-s + (0.892 + 0.239i)17-s + (−0.273 − 0.157i)19-s + (1.70 − 0.455i)23-s + (0.662 − 0.748i)25-s + 1.10·29-s + (0.893 + 1.54i)31-s + (−0.0957 + 0.995i)35-s + (−0.394 + 0.105i)37-s − 0.184i·41-s + (0.500 − 0.500i)43-s + (−0.0955 − 0.356i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.996 - 0.0892i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.996 - 0.0892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.362343807\)
\(L(\frac12)\) \(\approx\) \(1.362343807\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2.03 - 0.918i)T \)
7 \( 1 + (-1.31 + 2.29i)T \)
good11 \( 1 + (5.56 - 3.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.53 + 1.53i)T - 13iT^{2} \)
17 \( 1 + (-3.67 - 0.985i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.19 + 0.687i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-8.15 + 2.18i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.95T + 29T^{2} \)
31 \( 1 + (-4.97 - 8.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.39 - 0.642i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 1.18iT - 41T^{2} \)
43 \( 1 + (-3.28 + 3.28i)T - 43iT^{2} \)
47 \( 1 + (0.655 + 2.44i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.482 - 1.80i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.27 - 5.67i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.09 + 7.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.49 - 9.29i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.06iT - 71T^{2} \)
73 \( 1 + (-12.9 - 3.47i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-8.00 - 4.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.11 - 5.11i)T + 83iT^{2} \)
89 \( 1 + (-3.97 + 6.88i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.537 + 0.537i)T + 97iT^{2} \)
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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

−10.05037997266485809749916200761, −8.568331787133148044864191351530, −8.034895498910777179223242935564, −7.29043571340036208609186863540, −6.71217416020368252679677235014, −5.16583142240397075354021879226, −4.68757640912187080601612850085, −3.51172182494211510520113732258, −2.62528436815574587432725026986, −0.891471111330452992696372843776, 0.857487632386513337474819128832, 2.57240577514168079093268254107, 3.41332487800653919446180227876, 4.74363444708460077745015411681, 5.30537904708837309742400952703, 6.23030556488643862689183999952, 7.53462444851797919348815664427, 8.118354534800904411727267898874, 8.642444332265744593701552138189, 9.545356241093894940407884841260

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