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Properties

Label 1-2960-2960.1493-r0-0-0
Degree $1$
Conductor $2960$
Sign $0.924 - 0.380i$
Analytic cond. $13.7461$
Root an. cond. $13.7461$
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  + (0.939 − 0.342i)3-s + (0.984 + 0.173i)7-s + (0.766 − 0.642i)9-s + (−0.866 + 0.5i)11-s + (−0.642 + 0.766i)13-s + (0.766 − 0.642i)17-s + (0.939 − 0.342i)19-s + (0.984 − 0.173i)21-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)27-s + (−0.5 − 0.866i)29-s − i·31-s + (−0.642 + 0.766i)33-s + (−0.342 + 0.939i)39-s + (0.766 + 0.642i)41-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)3-s + (0.984 + 0.173i)7-s + (0.766 − 0.642i)9-s + (−0.866 + 0.5i)11-s + (−0.642 + 0.766i)13-s + (0.766 − 0.642i)17-s + (0.939 − 0.342i)19-s + (0.984 − 0.173i)21-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)27-s + (−0.5 − 0.866i)29-s − i·31-s + (−0.642 + 0.766i)33-s + (−0.342 + 0.939i)39-s + (0.766 + 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.924 - 0.380i$
Analytic conductor: \(13.7461\)
Root analytic conductor: \(13.7461\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (0:\ ),\ 0.924 - 0.380i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.737517888 - 0.5416335001i\)
\(L(\frac12)\) \(\approx\) \(2.737517888 - 0.5416335001i\)
\(L(1)\) \(\approx\) \(1.616922890 - 0.1687938553i\)
\(L(1)\) \(\approx\) \(1.616922890 - 0.1687938553i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (0.984 + 0.173i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.642 + 0.766i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - iT \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.984 - 0.173i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (0.984 - 0.173i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.190855912341114909801802071, −18.39662728573387207703381234318, −17.96080154202193392059433275689, −16.94532423795976645602097352720, −16.24295368977326550283790091526, −15.56345527401155649053166884698, −14.79572929068406253870677405979, −14.27809773010339102282295244355, −13.77220027250233722243770936269, −12.74892962283987216829514580511, −12.29193460098991443615934112123, −11.09199834528011576703588389049, −10.46622083797378674575886423056, −10.00783349303316225134802913333, −8.966948506857962720269103511702, −8.26726528247881832089769093347, −7.74861334497481529637598856113, −7.21936149990350589288476773984, −5.76692439407203691411105465059, −5.18103172544472502881744110007, −4.41912361978674489259814100753, −3.458033054552124701364471607648, −2.82927757776240815300591937140, −1.9374553217278881280257681927, −0.99910712436104172569408709365, 0.88581426125930908020142842234, 1.998042138278793910425439937819, 2.379718783789827260336981851712, 3.39428965008648857747999415151, 4.33420764769514084747993550310, 5.040482936631072746534749686370, 5.86556083190581065238050565439, 7.09924729394561262367822400373, 7.71377312138428121289188055435, 7.91663140994777262591573349344, 9.17215487586196700693697117438, 9.53676353445089419766459114216, 10.33588518663474663374640735889, 11.51902552433549986165313635226, 11.89404722347937444558745768671, 12.77064996646339134390343062288, 13.63932020595953037860550574644, 14.06111941214745924593199179957, 14.79805335572185225204621993290, 15.40830307064085281175809404793, 16.05605119895885112302276966873, 17.1295729923215776475875283311, 17.77837254104664208617546093406, 18.542829504473429685139519218607, 18.84996949837104433273115447813

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