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Properties

Label 1-3233-3233.3232-r0-0-0
Degree $1$
Conductor $3233$
Sign $1$
Analytic cond. $15.0139$
Root an. cond. $15.0139$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s − 14-s + 15-s + 16-s − 17-s + 18-s − 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s − 14-s + 15-s + 16-s − 17-s + 18-s − 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3233 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3233 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3233\)    =    \(53 \cdot 61\)
Sign: $1$
Analytic conductor: \(15.0139\)
Root analytic conductor: \(15.0139\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3233} (3232, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 3233,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.272031385\)
\(L(\frac12)\) \(\approx\) \(1.272031385\)
\(L(1)\) \(\approx\) \(1.025122746\)
\(L(1)\) \(\approx\) \(1.025122746\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad53 \( 1 \)
61 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
59 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−18.85329101910780626650875627490, −18.49815517897446027825020594207, −17.172546124037730660821003645972, −16.70011286273984704632775809155, −15.88137974069734585000890982243, −15.44982842035907941414136160277, −15.20457108920366114996349932921, −13.726940209342229096952275022934, −13.0824791493807308568329745449, −12.75103590674146542536281445329, −11.97649351156278990778900195011, −11.21026713194306401087070609661, −10.7580275753932779071591154980, −10.17634369566697601579022551186, −8.87448557738552060306673591657, −8.00667766231897435029303330315, −6.97563673417182214218788143313, −6.70367755394557418991304674480, −5.87583109060376824696091174650, −5.07710839551294418579207285121, −4.37116717168049897488793439737, −3.67341870848501692437705994512, −2.949532909376557273764624091416, −1.8437548820029777477763349928, −0.55302733063220763856424379013, 0.55302733063220763856424379013, 1.8437548820029777477763349928, 2.949532909376557273764624091416, 3.67341870848501692437705994512, 4.37116717168049897488793439737, 5.07710839551294418579207285121, 5.87583109060376824696091174650, 6.70367755394557418991304674480, 6.97563673417182214218788143313, 8.00667766231897435029303330315, 8.87448557738552060306673591657, 10.17634369566697601579022551186, 10.7580275753932779071591154980, 11.21026713194306401087070609661, 11.97649351156278990778900195011, 12.75103590674146542536281445329, 13.0824791493807308568329745449, 13.726940209342229096952275022934, 15.20457108920366114996349932921, 15.44982842035907941414136160277, 15.88137974069734585000890982243, 16.70011286273984704632775809155, 17.172546124037730660821003645972, 18.49815517897446027825020594207, 18.85329101910780626650875627490

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